Flashcards

Graphs made with Desmos. Road to 800 is not affiliated with Desmos.

Road to 800 · Desmos flashcards. Front of each card above the dashed line, back below. Graphs made with Desmos.

• Toolkit

  Solve for $t$.

  $4(t - 3) = t + 9$

  Road to 800

  Show the answer

  Answer

  $t = 7$

  Type in Desmos

  $4\left(x-3\right)=x+9$

  Where to look

  Click where the vertical line meets the $x$-axis: $(7, 0)$.

  

  Graph made with Desmos

  Tip Desmos only solves for $x$ and $y$, so type $t$ as $x$.

• Toolkit

  Solve. How many solutions are there?

  $x^2 - 3x = 10$

  Road to 800

  Show the answer

  Answer

  Two: $x = -2$ and $x = 5$

  Type in Desmos

  $x^{2}-3x=10$

  Where to look

  Each vertical line is one solution. Click each one on the $x$-axis.

  

  Graph made with Desmos

  Tip No line at all means no real solution.

• Toolkit

  Find the slope of the line through these points. Enter it as a fraction.

  $(2, 5)$ and $(8, 9)$

  Road to 800

  Show the answer

  Answer

  $\dfrac{2}{3}$

  Type in Desmos

  $\dfrac{\left(9-5\right)}{\left(8-2\right)}$

  Where to look

  Click the fraction button at the left of the row.

  

  Graph made with Desmos

  Tip As a decimal, .6667 is right but .67 is marked wrong.

• Toolkit

  Find $f(-1)$, $f(2)$ and $f(4)$.

  $f(x) = 2x^2 - 5x + 3$

  Road to 800

  Show the answer

  Answer

  $10$, $1$ and $15$

  Type in Desmos

  $f\left(x\right)=2x^{2}-5x+3$$f\left(\left[-1,2,4\right]\right)$

  Where to look

  Row 2 shows all three answers, in the same order.

  

  Graph made with Desmos

  Tip Start with f(x)=, not y=, so Desmos knows $f$.

• Toolkit

  Find the greatest value of $f(x)$ for $0 \le x \le 3$.

  $f(x) = -x^2 + 8x$

  Road to 800

  Show the answer

  Answer

  $15$

  Type in Desmos

  $f\left(x\right)=-x^{2}+8x\left\{0\le x\le3\right\}$$\left(3,f\left(3\right)\right)$

  Where to look

  The curve still rises at $x = 3$. Click the end dot: $(3, 15)$.

  

  Graph made with Desmos

  Tip The full curve’s top, $(4, 16)$, is outside $0$ to $3$.

• Toolkit

  For what value of $k$ does the line pass through $(3, 8)$?

  $y = kx - 4$

  Road to 800

  Show the answer

  Answer

  $k = 4$

  Type in Desmos

  $y=kx-4$$\left(3,8\right)$

  Where to look

  Drag the $k$ slider until the line goes through the dot.

  

  Graph made with Desmos

  Tip No slider yet? Click the “k” button under the row.

• Toolkit

  Which line passes through $(-1, 5)$ and $(2, -1)$?

  A) $2x + y = 3$B) $x + 2y = 9$C) $2x - y = -7$

  Road to 800

  Show the answer

  Answer

  A) $2x + y = 3$

  Type in Desmos

  $2x+y=3$$x+2y=9$$2x-y=-7$$\left[\left(-1,5\right),\left(2,-1\right)\right]$

  Where to look

  The green line goes through both dots.

  

  Graph made with Desmos

  Tip Each wrong line hits only one of the dots.

• Toolkit

  Do these lines ever cross? If so, where?

  $y = 0.5x + 2$$y = 0.45x + 4$

  Road to 800

  Show the answer

  Answer

  Yes, at $(40, 22)$

  Type in Desmos

  $y=0.5x+2$$y=0.45x+4$

  Where to look

  They look parallel. Zoom out until they cross, then click there.

  

  Graph made with Desmos

  Tip Different slopes always cross somewhere.

• Linear equations

  Solve.

  $\dfrac{x + 7}{4} = \dfrac{2x - 1}{3}$

  Road to 800

  Show the answer

  Answer

  $x = 5$

  Type in Desmos

  $\dfrac{\left(x+7\right)}{4}=\dfrac{\left(2x-1\right)}{3}$Keys: (x+7)/4 → =(2x-1)/3

  Where to look

  Click where the vertical line meets the $x$-axis: $(5, 0)$.

  

  Graph made with Desmos

  Tip → is the right-arrow key. It gets you out of the fraction.

• Linear equations

  What is the value of $2x + 1$?

  $5(x - 1) = 2x + 13$

  Road to 800

  Show the answer

  Answer

  $2x + 1 = 13$

  Type in Desmos

  $5\left(x-1\right)=2x+13$$y=2x+1$

  Where to look

  Click where the two lines cross: $(6, 13)$. Read the $y$-value.

  

  Graph made with Desmos

  Tip Don’t answer $6$. That’s $x$, not $2x + 1$.

• Linear equations

  This equation has the solution $x = 8$. $k$ is a constant. What is $k$?

  $5(x - k) = 2x + 3k$

  Road to 800

  Show the answer

  Answer

  $k = 3$

  Type in Desmos

  $5\left(8-x\right)=2\left(8\right)+3x$

  Where to look

  Click where the vertical line meets the $x$-axis: $(3, 0)$.

  

  Graph made with Desmos

  Tip Put $8$ in for $x$ first. Then type $k$ as $x$.

• Lines and linear functions

  $f$ is a linear function. Find $f(12)$.

  $f(2) = 1$, $f(5) = 10$, $f(9) = 22$

  Road to 800

  Show the answer

  Answer

  $f(12) = 31$

  Type in Desmos

  $x_{1}=\left[2,5,9\right]$$y_{1}=\left[1,10,22\right]$$y_{1}\sim mx_{1}+b$$12m+b$

  Where to look

  Row 3 shows $m = 3$ and $b = -5$. Row 4 shows $f(12)$.

  

  Graph made with Desmos

  Tip Delete any old $m$ or $b$ rows first.

• Lines and linear functions

  Find the $y$-intercept of $g$.

  $f(x) = 3x + 1$$g(x) = f(x + 2) - 5$

  Road to 800

  Show the answer

  Answer

  $(0, 2)$

  Type in Desmos

  $f\left(x\right)=3x+1$$g\left(x\right)=f\left(x+2\right)-5$$g\left(0\right)$

  Where to look

  Row 3 shows $g(0) = 2$. That’s where $g$ crosses the $y$-axis.

  

  Graph made with Desmos

• Lines and linear functions

  If $g(a) = 27$, what is the value of $a$?

  $g(x) = 2.5x - 8$

  Road to 800

  Show the answer

  Answer

  $a = 14$

  Type in Desmos

  $g\left(x\right)=2.5x-8$$27=g\left(x\right)$

  Where to look

  Row 2 draws a vertical line. Click it on the $x$-axis: $(14, 0)$.

  

  Graph made with Desmos

  Tip Put the $27$ first. g(x)=27 draws a flat line instead.

• Lines and linear functions

  $g$ is $f$ reflected across the $y$-axis. Where does the graph of $g$ cross the $x$-axis?

  $f(x) = 0.75x - 6$

  Road to 800

  Show the answer

  Answer

  $(-8, 0)$

  Type in Desmos

  $f\left(x\right)=0.75x-6$$g\left(x\right)=f\left(-x\right)$

  Where to look

  Click where $g$ crosses the $x$-axis: $(-8, 0)$.

  

  Graph made with Desmos

  Tip Across the $x$-axis would be $-f(x)$ instead.

• Lines and linear functions

  $f$ is a linear function. Find $f(10)$.

  $f(3) = 2f(1)$$f(4) = 15$

  Road to 800

  Show the answer

  Answer

  $f(10) = 33$

  Type in Desmos

  $f\left(x\right)=mx+b$$\left[f\left(3\right),f\left(4\right)\right]\sim\left[2f\left(1\right),15\right]$$f\left(10\right)$

  Where to look

  Row 2 finds $m = 3$ and $b = 3$. Row 3 shows $f(10)$.

  

  Graph made with Desmos

  Tip Don’t add sliders for $m$ and $b$. The ~ finds them.

• Lines and linear functions

  Moving this line $4$ units right has the same result as moving it $d$ units down. What is $d$?

  $y = 1.25x + 2$

  Road to 800

  Show the answer

  Answer

  $d = 5$

  Type in Desmos

  $f\left(x\right)=1.25x+2$$y=f\left(x-4\right)$$y=f\left(x\right)-d$

  Where to look

  Drag the $d$ slider until the two new lines become one: $d = 5$.

  

  Graph made with Desmos

  Tip Right $4$ goes inside: $f(x - 4)$. Down $d$ goes outside.

• Lines and linear functions

  $(a, b)$ is on the graph of the linear function $f$. Which point must be on $y = f(x + 4) - 3$?

  A) $(a + 4, b - 3)$B) $(a - 4, b - 3)$C) $(a - 4, b + 3)$

  Road to 800

  Show the answer

  Answer

  B) $(a - 4, b - 3)$

  Type in Desmos

  $f\left(x\right)=0.5x+2$$y=f\left(x+4\right)-3$$\left[\left(6,0\right),\left(-2,0\right),\left(-2,6\right)\right]$

  Where to look

  Dots: A, B, C. Only B, $(-2, 0)$, sits on the new line.

  

  Graph made with Desmos

  Tip Any line works for $f$. Here $(a, b) = (2, 3)$.

• Lines and linear functions

  Find the equation of the line through these points.

  $(-3, 2)$ and $(5, 6)$

  Road to 800

  Show the answer

  Answer

  $y = 0.5x + 3.5$

  Type in Desmos

  $x_{1}=\left[-3,5\right]$$y_{1}=\left[2,6\right]$$y_{1}\sim mx_{1}+b$

  Where to look

  Row 3 shows $m = 0.5$ and $b = 3.5$.

  

  Graph made with Desmos

  Tip List the $x$’s and the $y$’s in the same order.

• Lines and linear functions

  Where does this line cross the $y$-axis?

  slope $-2$, through $(4, 1)$

  Road to 800

  Show the answer

  Answer

  $(0, 9)$

  Type in Desmos

  $y-1=-2\left(x-4\right)$

  Where to look

  Click the line, then the dot on the $y$-axis.

  

  Graph made with Desmos

  Tip The point $(4, 1)$ becomes $x - 4$ and $y - 1$.

• Lines and linear functions

  A line through $(2, 7)$ is perpendicular to this line. Where does it cross the $y$-axis?

  $2x - 3y = 6$

  Road to 800

  Show the answer

  Answer

  $(0, 10)$

  Type in Desmos

  $2x-3y=6$$y-7=-1.5\left(x-2\right)$

  Where to look

  Click the new line, then its dot on the $y$-axis.

  

  Graph made with Desmos

  Tip The given line has slope $\dfrac{2}{3}$, so use $-\dfrac{3}{2} = -1.5$.

• Lines and linear functions

  Find the slope.

  $3x - 4y = 24$

  Road to 800

  Show the answer

  Answer

  Slope $\dfrac{3}{4}$

  Type in Desmos

  $3x-4y=24$

  Where to look

  Click the line. Its dots on the axes are $(8, 0)$ and $(0, -6)$.

  

  Graph made with Desmos

  Tip Rise $6$ over run $8$ is $\dfrac{3}{4}$.

• Lines and linear functions

  These three points are on one line. Find $a$.

  $(1, 5)$, $(4, 14)$, $(a, 26)$

  Road to 800

  Show the answer

  Answer

  $a = 8$

  Type in Desmos

  $x_{1}=\left[1,4\right]$$y_{1}=\left[5,14\right]$$y_{1}\sim mx_{1}+b$$y=26$

  Where to look

  Click where the fitted line meets $y = 26$.

  

  Graph made with Desmos

• Lines and linear functions

  Which equation is the same line as $10x + 4y = 60$?

  A) $y = 15 - 2.5x$B) $y = 15 - 0.4x$C) $y = 6 - 2.5x$

  Road to 800

  Show the answer

  Answer

  A) $y = 15 - 2.5x$

  Type in Desmos

  $10x+4y=60$$y=15-2.5x$$y=15-0.4x$$y=6-2.5x$

  Where to look

  Only A’s line lies exactly on top of the first line.

  

  Graph made with Desmos

  Tip B crosses the first line and C runs beside it.

• Lines and linear functions

  These three points lie on one line. Find $4a + b$.

  $(2, a)$, $(4, -3)$, $(12, b)$

  Road to 800

  Show the answer

  Answer

  $4a + b = -15$

  Type in Desmos

  $m=1$$f\left(x\right)=m\left(x-4\right)-3$$4f\left(2\right)+f\left(12\right)$

  Where to look

  Row 3 stays at $-15$ however you drag $m$.

  

  Graph made with Desmos

  Tip $a$ is $f(2)$ and $b$ is $f(12)$: both points are on the line.

• Systems of equations and inequalities

  How many solutions does this equation have?

  $2(3x - 1) = 6x + 5$

  Road to 800

  Show the answer

  Answer

  No solution

  Type in Desmos

  $y=2\left(3x-1\right)$$y=6x+5$

  Where to look

  Graph each side. The lines are parallel, so they never meet.

  

  Graph made with Desmos

  Tip One line on top of the other means infinitely many.

• Systems of equations and inequalities

  $k$ is a constant, and this equation has no solution. What is $k$?

  $3(x + 4) - 5x = kx - 7$

  Road to 800

  Show the answer

  Answer

  $k = -2$

  Type in Desmos

  $y=3\left(x+4\right)-5x$$y=kx-7$

  Where to look

  Drag the $k$ slider until the two lines are parallel: $k = -2$.

  

  Graph made with Desmos

  Tip No slider yet? Click the “k” button under the row.

• Systems of equations and inequalities

  Solve the system.

  $3x + 2y = 16$$x - y = 2$

  Road to 800

  Show the answer

  Answer

  $x = 4$, $y = 2$

  Type in Desmos

  $3x+2y=16$$x-y=2$

  Where to look

  Click where the lines cross.

  

  Graph made with Desmos

• Systems of equations and inequalities

  For what value of $k$ does the system have no solution?

  $kx - 2y = 3$$6x - 4y = -10$

  Road to 800

  Show the answer

  Answer

  $k = 3$

  Type in Desmos

  $kx-2y=3$$6x-4y=-10$

  Where to look

  Drag the $k$ slider until the lines are parallel.

  

  Graph made with Desmos

  Tip No slider yet? Click the “k” button under the row.

• Systems of equations and inequalities

  How many solutions does this system have?

  $6x - 4y = 10$$9x - 6y = 15$

  Road to 800

  Show the answer

  Answer

  Infinitely many

  Type in Desmos

  $6x-4y=10$$9x-6y=15$

  Where to look

  Only one line shows: row 2 lies on top of row 1.

  

  Graph made with Desmos

  Tip Two separate parallel lines would mean no solution.

• Systems of equations and inequalities

  The solution $(x, y)$ of this system has $y = 6$. Find $k$.

  $4x - 3y = 2$$kx + 2y = 22$

  Road to 800

  Show the answer

  Answer

  $k = 2$

  Type in Desmos

  $4x-3y=2$$kx+2y=22$$y=6$

  Where to look

  Drag $k$ until all three lines meet at one point: $(5, 6)$.

  

  Graph made with Desmos

  Tip No slider yet? Click the “k” button under the row.

• Systems of equations and inequalities

  For what value of $c$ does the system have infinitely many solutions?

  $3x + 4y = 7$$-9x - 12y = c$

  Road to 800

  Show the answer

  Answer

  $c = -21$

  Type in Desmos

  $3x+4y=7$$-9x-12y=c$

  Where to look

  Drag $c$ until the two lines become one.

  

  Graph made with Desmos

  Tip The slider stops at $-10$: click that end and type $-30$.

• Systems of equations and inequalities

  The solution of this system is $x = a$ and $y = b$. Find $a$ and $b$.

  $x + y = 2a - 1$$x - 2y = b - 11$

  Road to 800

  Show the answer

  Answer

  $a = 7$, $b = 6$

  Type in Desmos

  $x+y=2x-1$$x-2y=y-11$

  Where to look

  Click where the lines cross: $(7, 6)$ is $(a, b)$.

  

  Graph made with Desmos

  Tip Type $x$ in place of $a$, and $y$ in place of $b$.

• Systems of equations and inequalities

  What is the greatest whole number that makes this true?

  $6x - 5 \le 2x + 21$

  Road to 800

  Show the answer

  Answer

  $6$

  Type in Desmos

  $6x-5\le2x+21$

  Where to look

  Click the edge of the shading: $(6.5, 0)$.

  

  Graph made with Desmos

  Tip $6.5$ isn’t a whole number, so go down to $6$.

• Systems of equations and inequalities

  Which point is a solution of both?

  $y > x - 2$$y \le -2x + 7$$(3, 2)$ $(1, 4)$ $(2, 0)$

  Road to 800

  Show the answer

  Answer

  $(1, 4)$

  Type in Desmos

  $y>x-2$$y\le-2x+7$$\left[\left(3,2\right),\left(1,4\right),\left(2,0\right)\right]$

  Where to look

  Only one dot sits where the two shadings overlap.

  

  Graph made with Desmos

  Tip A dot on a dashed edge does not count.

• Systems of equations and inequalities

  What is the greatest possible value of $y$?

  $y \ge 0$$2x + y \le 12$$y \le x + 3$

  Road to 800

  Show the answer

  Answer

  $y = 6$, at $(3, 6)$

  Type in Desmos

  $y\ge0$$2x+y\le12$$y\le x+3$

  Where to look

  Click the top corner of the darkest region.

  

  Graph made with Desmos

  Tip The largest or smallest value is always at a corner.

• Systems of equations and inequalities

  Solve.

  $-2 < 4 - 3x \le 13$

  Road to 800

  Show the answer

  Answer

  $-3 \le x < 2$

  Type in Desmos

  $-2<4-3x$$4-3x\le13$

  Where to look

  The overlap runs from $x = -3$ (solid edge) to $x = 2$ (dashed edge).

  

  Graph made with Desmos

  Tip Desmos can’t graph this double inequality in one row, so split it.

• Systems of equations and inequalities

  The point $(2, y)$ is a solution of both. How many integers can $y$ be?

  $y > x - 1$$y \le -2x + 11$

  Road to 800

  Show the answer

  Answer

  $6$ integers: $2$ to $7$

  Type in Desmos

  $y>x-1$$y\le-2x+11$$x=2$

  Where to look

  $x = 2$ crosses the overlap from $(2, 1)$ up to $(2, 7)$.

  

  Graph made with Desmos

  Tip $(2, 1)$ is on the dashed edge, so $y = 1$ doesn’t count.

• Systems of equations and inequalities

  If $(x, y)$ is a solution of both, what is the greatest possible value of $x + y$?

  $2x + 5y \le 30$$4x + y \le 24$

  Road to 800

  Show the answer

  Answer

  $x + y = 9$, at $(5, 4)$

  Type in Desmos

  $2x+5y\le30$$4x+y\le24$$x+y=k$

  Where to look

  Drag $k$ up until the line last touches the overlap, at $(5, 4)$.

  

  Graph made with Desmos

  Tip Any higher $k$ and the line misses the overlap.

• Linear word problems

  A car loses the same value each year. Find $V(t)$, its value in dollars $t$ years after 2021.

  2019: $\$21{,}500$2024: $\$14{,}000$

  Road to 800

  Show the answer

  Answer

  $V(t) = -1500t + 18500$

  Type in Desmos

  $x_{1}=\left[2019,2024\right]-2021$$y_{1}=\left[21500,14000\right]$$y_{1}\sim mx_{1}+b$

  Where to look

  Row 3 shows $m = -1500$ and $b = 18500$.

  

  Graph made with Desmos

  Tip Subtracting 2021 turns each year into $t$: 2019 becomes $-2$.

• Linear word problems

  A plan costs $\$30$ a month for up to $5$ GB, plus $\$8$ per GB over $5$. Which gives $C(g)$ for $g \ge 5$ GB?

  A) $C(g) = 8g + 30$B) $C(g) = 8g - 10$C) $C(g) = 8g - 5$

  Road to 800

  Show the answer

  Answer

  B) $C(g) = 8g - 10$

  Type in Desmos

  $A\left(x\right)=8x+30$$B\left(x\right)=8x-10$$C\left(x\right)=8x-5$$\left[A\left(7\right),B\left(7\right),C\left(7\right)\right]$

  Where to look

  By hand, $7$ GB costs $30 + 2 \cdot 8 = \$46$. Only $B(7)$ is $46$.

  

  Graph made with Desmos

  Tip Any $g$ over $5$ works. Pick one that is easy by hand.

• Linear word problems

  A plant is $H(w)$ cm tall $w$ weeks after planting, for $w \ge 4$. What does the $20$ mean?

  $H(w) = 1.5(w - 4) + 20$

  Road to 800

  Show the answer

  Answer

  The height at week 4: $20$ cm

  Type in Desmos

  $H\left(w\right)=1.5\left(w-4\right)+20$$H\left(4\right)$$H\left(0\right)$

  Where to look

  Row 2 shows $H(4) = 20$. Row 3 shows $H(0) = 14$, not $20$.

  

  Graph made with Desmos

  Tip Desmos accepts H(w)= as well as H(x)=.

• Linear word problems

  When do the two plans cost the same? What is that cost?

  A: $\$30 + \$12$ per monthB: $\$75 + \$7$ per month

  Road to 800

  Show the answer

  Answer

  After $9$ months, $\$138$ each

  Type in Desmos

  $y=30+12x$$y=75+7x$

  Where to look

  Zoom out, then click where the lines cross.

  

  Graph made with Desmos

  Tip $x$ is the number of months; $y$ is the cost.

• Linear word problems

  A shop sold $70$ plants, all ferns or cacti, for $\$470$. How many were ferns?

  ferns: $\$9$ eachcacti: $\$5$ each

  Road to 800

  Show the answer

  Answer

  $30$ ferns

  Type in Desmos

  $x+y=70$$9x+5y=470$

  Where to look

  Click where the lines cross: $(30, 40)$.

  

  Graph made with Desmos

  Tip $x$ counts the ferns and $y$ counts the cacti.

• Linear word problems

  Solutions A and B are mixed to make the batch. How many liters of B are used?

  A: $20\%$ saltB: $45\%$ saltbatch: $50$ liters, $30\%$ salt

  Road to 800

  Show the answer

  Answer

  $20$ liters of B

  Type in Desmos

  $x+y=50$$0.2x+0.45y=0.3\cdot50$

  Where to look

  Click where the lines cross: $(30, 20)$. $y$ is the liters of B.

  

  Graph made with Desmos

  Tip Type each percent as a decimal: $45\%$ is $0.45$.

• Polynomial functions

  Find $f(g(-2))$.

  $f(x) = x^2 - 3x$$g(x) = 2x + 1$

  Road to 800

  Show the answer

  Answer

  $18$

  Type in Desmos

  $f\left(x\right)=x^{2}-3x$$g\left(x\right)=2x+1$$f\left(g\left(-2\right)\right)$

  Where to look

  Read the number next to the last row.

  

  Graph made with Desmos

• Polynomial functions

  Find all the zeros of $p$.

  $p(x) = x^3 - 3x^2 - 9x + 27$

  Road to 800

  Show the answer

  Answer

  $x = -3$ and $x = 3$

  Type in Desmos

  $y=x^{3}-3x^{2}-9x+27$

  Where to look

  Click where the curve meets the $x$-axis: $(-3, 0)$ and $(3, 0)$.

  

  Graph made with Desmos

  Tip At $x = 3$ it just touches the axis. That still counts.

• Polynomial functions

  Find the remainder when $p(x)$ is divided by $x - 3$.

  $p(x) = x^3 - 4x^2 + 2x + 9$

  Road to 800

  Show the answer

  Answer

  $6$

  Type in Desmos

  $p\left(x\right)=x^{3}-4x^{2}+2x+9$$p\left(3\right)$

  Where to look

  The number next to $p(3)$ is the remainder.

  

  Graph made with Desmos

  Tip Dividing by $x + 3$ instead? Then type $p(-3)$.

• Polynomial functions

  Find the zeros of $y = f(x + 3)$.

  $f(x) = x^2 - 2x - 8$

  Road to 800

  Show the answer

  Answer

  $x = -5$ and $x = 1$

  Type in Desmos

  $f\left(x\right)=x^{2}-2x-8$$y=f\left(x+3\right)$

  Where to look

  Click where the new curve crosses the $x$-axis: $(-5, 0)$ and $(1, 0)$.

  

  Graph made with Desmos

• Polynomial functions

  For which values of $x$ is this undefined?

  $\dfrac{x^2 - 9}{x^2 + x - 6}$

  Road to 800

  Show the answer

  Answer

  $x = -3$ and $x = 2$

  Type in Desmos

  $y=x^{2}+x-6$

  Where to look

  Graph only the bottom. Click its $x$-intercepts: $(-3, 0)$ and $(2, 0)$.

  

  Graph made with Desmos

  Tip Don’t graph the whole fraction: Desmos hides the hole at $-3$.

• Quadratics and parabolas

  What is the maximum value of $f$?

  $f(x) = -3x^2 + 18x - 5$

  Road to 800

  Show the answer

  Answer

  $22$

  Type in Desmos

  $y=-3x^{2}+18x-5$

  Where to look

  Click the top point: $(3, 22)$.

  

  Graph made with Desmos

  Tip The maximum is the $y$-value, $22$, not $x = 3$.

• Quadratics and parabolas

  Find the equation of the parabola through these points.

  $(0, 3)$, $(2, -1)$, $(5, 8)$

  Road to 800

  Show the answer

  Answer

  $y = x^2 - 4x + 3$

  Type in Desmos

  $x_{1}=\left[0,2,5\right]$$y_{1}=\left[3,-1,8\right]$$y_{1}\sim ax_{1}^{2}+bx_{1}+c$

  Where to look

  Read $a$, $b$ and $c$ under the last row.

  

  Graph made with Desmos

  Tip The ~ finds the parabola that fits the points.

• Quadratics and parabolas

  For how many seconds is the ball above $53$ feet?

  $h = -16t^2 + 64t + 5$

  Road to 800

  Show the answer

  Answer

  $2$ seconds

  Type in Desmos

  $y=-16x^{2}+64x+5$$y=53$

  Where to look

  Click the two crossings: $(1, 53)$ and $(3, 53)$.

  

  Graph made with Desmos

  Tip Type $t$ as $x$. The answer is the gap: $3 - 1$.

• Quadratics and parabolas

  Which is a solution?

  $x^2 - 6x + 4 = 0$A) $3 + \sqrt{5}$B) $3 + \sqrt{13}$

  Road to 800

  Show the answer

  Answer

  A) $3 + \sqrt{5}$

  Type in Desmos

  $y=x^{2}-6x+4$$x=3+\sqrt{5}$$x=3+\sqrt{13}$

  Where to look

  Only the line $x = 3 + \sqrt{5}$ meets the curve on the $x$-axis.

  

  Graph made with Desmos

• Quadratics and parabolas

  How many real solutions does this have?

  $4x^2 + 9 = 12x$

  Road to 800

  Show the answer

  Answer

  Exactly one

  Type in Desmos

  $4x^{2}+9=12x$

  Where to look

  Only one vertical line shows up, at $x = 1.5$.

  

  Graph made with Desmos

  Tip No lines at all would mean no real solutions.

• Quadratics and parabolas

  The graphs meet at exactly one point. $k$ is positive. Find $k$.

  $y = kx$ and $y = x^2 + 9$

  Road to 800

  Show the answer

  Answer

  $k = 6$

  Type in Desmos

  $y=\dfrac{x^{2}+9}{x}$

  Where to look

  Click the low point on the right: $(3, 6)$. Its height is $k$.

  

  Graph made with Desmos

  Tip Get $k$ alone first: $k = \dfrac{x^2 + 9}{x}$.

• Quadratics and parabolas

  What is the sum of the solutions?

  $2x^2 = 10x - 7$

  Road to 800

  Show the answer

  Answer

  $5$

  Type in Desmos

  $2x^{2}=10x-7$

  Where to look

  Click both vertical lines: $x \approx 0.842$ and $x \approx 4.158$. Add them.

  

  Graph made with Desmos

• Quadratics and parabolas

  Solve for $x$.

  $x - 7\sqrt{x} + 10 = 0$

  Road to 800

  Show the answer

  Answer

  $x = 4$ or $x = 25$

  Type in Desmos

  $y=x+10-7\sqrt{x}$

  Where to look

  Click the $x$-intercepts: $(4, 0)$ and $(25, 0)$.

  

  Graph made with Desmos

  Tip Type √ last, or the + 10 lands inside it.

• Quadratics and parabolas

  Where do the two graphs meet?

  $y = x^2 - 4x + 1$$2x - y = 4$

  Road to 800

  Show the answer

  Answer

  $(1, -2)$ and $(5, 6)$

  Type in Desmos

  $y=x^{2}-4x+1$$2x-y=4$

  Where to look

  Click the two crossings: $(1, -2)$ and $(5, 6)$.

  

  Graph made with Desmos

  Tip Type the line as it is; no need to solve for $y$.

• Quadratics and parabolas

  The graphs meet at two points. $k$ is an integer. What is the least value of $k$?

  $y = x^2 - 3x$$y = x + k$

  Road to 800

  Show the answer

  Answer

  $k = -3$

  Type in Desmos

  $y=x^{2}-3x$$y=x+k$

  Where to look

  Drag $k$. At $k = -3$ the line crosses twice; at $-4$ it only touches.

  

  Graph made with Desmos

  Tip No slider yet? Click the “k” button under the row.

• Quadratics and parabolas

  $(x, y)$ is a solution of the system, and $x > 0$. What is $y$?

  $xy = 8$$x - y = 2$

  Road to 800

  Show the answer

  Answer

  $y = 2$

  Type in Desmos

  $xy=8$$x-y=2$

  Where to look

  Click the crossing on the right: $(4, 2)$.

  

  Graph made with Desmos

  Tip Desmos graphs $xy = 8$ as it is. No need to solve for $y$.

• Quadratics and parabolas

  This is true for all $x$. Find $a$ and $b$.

  $(x + a)(x + 2a) = x^2 + 9x + b$

  Road to 800

  Show the answer

  Answer

  $a = 3$, $b = 18$

  Type in Desmos

  $x_{1}=\left[1,2,3,4\right]$$\left(x_{1}+a\right)\left(x_{1}+2a\right)\sim x_{1}^{2}+9x_{1}+b$

  Where to look

  Read $a$ and $b$ under the last row.

  

  Graph made with Desmos

• Quadratics and parabolas

  This is true for all $x$. Find $h$ and $k$.

  $2x^2 - 12x + 23 = 2(x - h)^2 + k$

  Road to 800

  Show the answer

  Answer

  $h = 3$, $k = 5$

  Type in Desmos

  $y=2x^{2}-12x+23$

  Where to look

  Click the lowest point: $(3, 5)$. That point is $(h, k)$.

  

  Graph made with Desmos

• Absolute value

  Which values of $x$ make this true?

  $|x - 24| \le 6$

  Road to 800

  Show the answer

  Answer

  $18 \le x \le 30$

  Type in Desmos

  $\left|x-24\right|\le6$

  Where to look

  The shading runs from $x = 18$ to $x = 30$.

  

  Graph made with Desmos

  Tip The middle is $24$, and each edge is $6$ away.

• Absolute value

  How many solutions does this have?

  $|x^2 - 6x| = 5$

  Road to 800

  Show the answer

  Answer

  $4$ solutions

  Type in Desmos

  $y=\operatorname{abs}\left(x^{2}-6x\right)$$y=5$

  Where to look

  Count where the line $y = 5$ crosses the curve: $4$ times.

  

  Graph made with Desmos

• Absolute value

  Solve for $x$.

  $|x - 5| = 3x + 1$

  Road to 800

  Show the answer

  Answer

  $x = 1$

  Type in Desmos

  $y=\operatorname{abs}\left(x-5\right)$$y=3x+1$

  Where to look

  Click where they cross: $(1, 4)$. They cross only once.

  

  Graph made with Desmos

  Tip Algebra also gives $x = -3$, but it does not work.

• Ratios, proportions and rates

  How many cm on the map stand for $72$ km?

  Map scale: $2$ cm $= 15$ km

  Road to 800

  Show the answer

  Answer

  $9.6$ cm

  Type in Desmos

  $\dfrac{2}{15}=\dfrac{x}{72}$Keys: 2/15 → =x/72

  Where to look

  Click where the vertical line meets the $x$-axis: $(9.6, 0)$.

  

  Graph made with Desmos

  Tip → is the right-arrow key. It gets you out of the fraction.

• Ratios, proportions and rates

  A faucet drips $12$ mL a minute. How many liters is that per day?

  $1$ L $= 1000$ mL

  Road to 800

  Show the answer

  Answer

  $17.28$ liters

  Type in Desmos

  $12\cdot60\cdot\dfrac{24}{1000}$

  Where to look

  Read the answer on the right: $17.28$.

  

  Graph made with Desmos

  Tip $\times 60$ per hour, $\times 24$ per day, $\div 1000$ to get liters.

• Ratios, proportions and rates

  How long do both pipes take to fill the tank together?

  A alone: $4$ hoursB alone: $12$ hours

  Road to 800

  Show the answer

  Answer

  $3$ hours

  Type in Desmos

  $\dfrac{x}{4}+\dfrac{x}{12}=1$Keys: x/4 → +x/12 → =1

  Where to look

  Click where the vertical line meets the $x$-axis: $(3, 0)$.

  

  Graph made with Desmos

  Tip In $x$ hours, A fills $\dfrac{x}{4}$ of the tank, B fills $\dfrac{x}{12}$.

• Ratios, proportions and rates

  A car travels at $54$ km per hour. What is that in meters per second?

  $1$ km $= 1000$ m

  Road to 800

  Show the answer

  Answer

  $15$ meters per second

  Type in Desmos

  $54\cdot\dfrac{1000}{3600}$

  Where to look

  Read the answer on the right: $15$.

  

  Graph made with Desmos

  Tip $\times 1000$ for meters, $\div 3600$ for seconds.

• Ratios, proportions and rates

  A trip is $120$ miles at $60$ mph, then $120$ miles at $40$ mph. Find the average speed.

  Road to 800

  Show the answer

  Answer

  $48$ miles per hour

  Type in Desmos

  $\dfrac{240}{\dfrac{120}{60}+\dfrac{120}{40}}$Keys: 240/120/60 → +120/40

  Where to look

  Read the answer on the right: $48$.

  

  Graph made with Desmos

  Tip Total distance $\div$ total time. It is not the average of $60$ and $40$.

• Percentages

  Find the number.

  $63$ is $35\%$ of what number?

  Road to 800

  Show the answer

  Answer

  $180$

  Type in Desmos

  $63=35\%\operatorname{of}x$

  Where to look

  Click where the vertical line meets the $x$-axis: $(180, 0)$.

  

  Graph made with Desmos

  Tip Type it in order: “is” means $=$, “what” means $x$.

• Percentages

  What is the total percent change in the price?

  Up $30\%$, then down $10\%$

  Road to 800

  Show the answer

  Answer

  Up $17\%$

  Type in Desmos

  $1.3\cdot0.9$

  Where to look

  Read $1.17$ on the right: the new price is $117\%$ of the old.

  

  Graph made with Desmos

  Tip Up $30\%$ is $\times\,1.3$. Down $10\%$ is $\times\,0.9$.

• Percentages

  What was the coat’s original price?

  $20\%$ off, then $8\%$ taxFinal price: $\$97.20$

  Road to 800

  Show the answer

  Answer

  $\$112.50$

  Type in Desmos

  $x\cdot0.8\cdot1.08=97.2$

  Where to look

  Click where the vertical line meets the $x$-axis: $(112.5, 0)$.

  

  Graph made with Desmos

  Tip $20\%$ off is $\times\,0.8$. $8\%$ tax is $\times\,1.08$.

• Percentages

  How many liters of pure acid must be added?

  Start: $30$ L of $10\%$ acidGoal: $25\%$ acid

  Road to 800

  Show the answer

  Answer

  $6$ liters

  Type in Desmos

  $0.1\cdot30+x=0.25\left(30+x\right)$

  Where to look

  Click where the vertical line meets the $x$-axis: $(6, 0)$.

  

  Graph made with Desmos

  Tip Acid you have + acid you add = acid in the mix.

• Percentages

  What percent of $80$ is $18$?

  Road to 800

  Show the answer

  Answer

  $22.5\%$

  Type in Desmos

  $\dfrac{18}{80}\cdot100$Keys: 18/80 → *100

  Where to look

  Read the answer on the right: $22.5$.

  

  Graph made with Desmos

  Tip Part $\div$ whole, then $\times\,100$.

• Percentages

  A price rises from $\$64$ to $\$80$. Find the percent increase.

  Road to 800

  Show the answer

  Answer

  $25\%$

  Type in Desmos

  $\dfrac{\left(80-64\right)}{64}\cdot100$Keys: (80-64)/64 → *100

  Where to look

  Read the answer on the right: $25$.

  

  Graph made with Desmos

  Tip Divide by the old value, $64$, not by $80$.

• Exponents and exponential functions

  After how many whole years is $V$ first over $2{,}000$?

  $V = 1500(1.04)^t$

  Road to 800

  Show the answer

  Answer

  $8$ years

  Type in Desmos

  $y=1500\left(1.04\right)^{x}$$y=2000$

  Where to look

  Click where they cross: $x \approx 7.33$.

  

  Graph made with Desmos

  Tip At $7$ years $V$ is still under $2{,}000$, so round up.

• Exponents and exponential functions

  Find $a$ and $b$ if $y = a \cdot b^x$ goes through both points.

  $(2, 36)$ and $(4, 81)$

  Road to 800

  Show the answer

  Answer

  $a = 16$, $b = 1.5$

  Type in Desmos

  $x_{1}=\left[2,4\right]$$y_{1}=\left[36,81\right]$$y_{1}\sim ab^{x_{1}}$

  Where to look

  Read $a$ and $b$ under the last row.

  

  Graph made with Desmos

• Exponents and exponential functions

  A dose halves every $k$ hours. Find $k$.

  $800(0.5)^{\frac{15}{k}} = 100$

  Road to 800

  Show the answer

  Answer

  $k = 5$ hours

  Type in Desmos

  $y=800\left(0.5\right)^{\frac{15}{x}}$$y=100$

  Where to look

  Click where they cross: $(5, 100)$.

  

  Graph made with Desmos

  Tip Desmos only solves for $x$, so type $k$ as $x$.

• Exponents and exponential functions

  Find $b$ so this is true for all $t$.

  $800(1.44)^{\frac{t}{2}} = 800 \cdot b^t$

  Road to 800

  Show the answer

  Answer

  $b = 1.2$

  Type in Desmos

  $f\left(x\right)=800\left(1.44\right)^{\frac{x}{2}}$$\dfrac{f\left(1\right)}{f\left(0\right)}$

  Where to look

  The number next to the last row is $b$: $1.2$.

  

  Graph made with Desmos

• Exponents and exponential functions

  This is true for all $t > 0$. Find $k$.

  $\sqrt{t} \cdot \sqrt[3]{t} = t^k$

  Road to 800

  Show the answer

  Answer

  $k = \dfrac{5}{6}$

  Type in Desmos

  $y=2^{x}$

  $y=\sqrt{2}\sqrt[3]{2}$Keys: y=sqrt(2) → cbrt(2)

  Where to look

  Click where they cross: $x \approx 0.83333$, which is $\dfrac{5}{6}$.

  

  Graph made with Desmos

  Tip Put $2$ in for $t$, and let $x$ stand for $k$.

• Exponents and exponential functions

  Linear or exponential? Find the equation.

  $x$: $0, 1, 2, 3$$y$: $6, 9, 13.5, 20.25$

  Road to 800

  Show the answer

  Answer

  Exponential: $y = 6(1.5)^x$

  Type in Desmos

  $x_{1}=\left[0,1,2,3\right]$$y_{1}=\left[6,9,13.5,20.25\right]$$y_{1}\sim ab^{x_{1}}$

  Where to look

  Read $a = 6$ and $b = 1.5$.

  

  Graph made with Desmos

  Tip Each $y$ is $1.5$ times the one before.

• Rewriting expressions and equations

  Solve for $n$.

  $P = \dfrac{5}{n + 2}$A) $\dfrac{5 - 2P}{P}$B) $\dfrac{5}{P - 2}$

  Road to 800

  Show the answer

  Answer

  A) $n = \dfrac{5 - 2P}{P}$

  Type in Desmos

  $n=8$$P=\dfrac{5}{n+2}$$\dfrac{5-2P}{P}$$\dfrac{5}{P-2}$

  Where to look

  Choice A gives back $8$, the $n$ you picked.

  

  Graph made with Desmos

  Tip Avoid $0$ and $1$: a wrong choice can match by luck.

• Rewriting expressions and equations

  Solve for $x$.

  $\sqrt{x + 7} = x + 1$

  Road to 800

  Show the answer

  Answer

  $x = 2$ only

  Type in Desmos

  $y=\sqrt{x+7}$$y=x+1$

  Where to look

  Click where they cross: $(2, 3)$. It is the only crossing.

  

  Graph made with Desmos

  Tip Algebra also gives $x = -3$, but it fails the check.

• Rewriting expressions and equations

  Solve for $x$.

  $\dfrac{4x}{x - 1} = \dfrac{4}{x - 1} + 1$

  Road to 800

  Show the answer

  Answer

  No solution

  Type in Desmos

  $y=\dfrac{4x}{x-1}$$y=1+\dfrac{4}{x-1}$

  Where to look

  Graph each side. The two curves never cross.

  

  Graph made with Desmos

  Tip Algebra gives $x = 1$, but that makes the bottom $0$.

• Rewriting expressions and equations

  Which is equal to this?

  $(2x + 3)^2 - (x - 1)^2$A) $(3x + 2)(x + 4)$B) $3x^2 + 8$

  Road to 800

  Show the answer

  Answer

  A) $(3x + 2)(x + 4)$

  Type in Desmos

  $y=\left(2x+3\right)^{2}-\left(x-1\right)^{2}$$y=\left(3x+2\right)\left(x+4\right)$$y=3x^{2}+8$

  Where to look

  Choice A draws right on top of the first curve. B does not.

  

  Graph made with Desmos

• Rewriting expressions and equations

  Which is a factor?

  $6x^2 + 7x - 20$A) $2x - 5$B) $3x - 4$

  Road to 800

  Show the answer

  Answer

  B) $3x - 4$

  Type in Desmos

  $y=6x^{2}+7x-20$

  Where to look

  Click the $x$-intercepts: $(-2.5, 0)$ and $(1.33333, 0)$.

  

  Graph made with Desmos

  Tip $x = \dfrac{4}{3}$ makes $3x - 4$ equal $0$.

• Rewriting expressions and equations

  This is true for all $x \ne -3$. Find $A$.

  $\dfrac{2x^2 + 7x - 1}{x + 3}$$= 2x + 1 + \dfrac{A}{x + 3}$

  Road to 800

  Show the answer

  Answer

  $A = -4$

  Type in Desmos

  $y=\dfrac{2x^{2}+7x-1}{x+3}$$y=2x+1+\dfrac{A}{x+3}$

  Where to look

  Drag $A$ until the second curve lies right on the first.

  

  Graph made with Desmos

  Tip No slider yet? Click the “A” button under the row.

• Rewriting expressions and equations

  Which is equal to this?

  $(a + 2b)(a - b) - (a - b)^2$A) $3ab - 3b^2$B) $-ab - b^2$

  Road to 800

  Show the answer

  Answer

  A) $3ab - 3b^2$

  Type in Desmos

  $a=2$$b=3$$\left(a+2b\right)\left(a-b\right)-\left(a-b\right)^{2}$$3ab-3b^{2}$

  Where to look

  Rows 3 and 4 both show $-9$. Choice B would give $-15$.

  

  Graph made with Desmos

  Tip Give $a$ and $b$ different values, not $0$ or $1$.

• Data and probability

  Find the mean and the median.

  $12, 15, 15, 18, 40$

  Road to 800

  Show the answer

  Answer

  Mean $20$, median $15$

  Type in Desmos

  $L=\left[12,15,15,18,40\right]$$\operatorname{mean}\left(L\right)$$\operatorname{median}\left(L\right)$

  Where to look

  Read the answers on the right.

  

  Graph made with Desmos

• Data and probability

  Find the mean quiz score.

  Score: $1, 2, 3, 4$Students: $2, 6, 7, 5$

  Road to 800

  Show the answer

  Answer

  Mean $2.75$

  Type in Desmos

  $V=\left[1,2,3,4\right]$$F=\left[2,6,7,5\right]$$\dfrac{\operatorname{total}\left(V\cdot F\right)}{\operatorname{total}\left(F\right)}$

  Where to look

  Read the answer next to the last row: $2.75$.

  

  Graph made with Desmos

  Tip Divide by all $20$ students, not by the $4$ scores.

• Data and probability

  Find the interquartile range (IQR).

  $1, 2, 4, 5, 7, 8, 10, 13$

  Road to 800

  Show the answer

  Answer

  IQR $= 6$

  Type in Desmos

  $L=\left[1,2,4,5,7,8,10,13\right]$$\operatorname{quartile}\left(L,1\right)$$\operatorname{quartile}\left(L,3\right)$

  Where to look

  Read Q1 $= 3$ and Q3 $= 9$, then subtract.

  

  Graph made with Desmos

• Data and probability

  Which set has the larger standard deviation?

  A: $10, 19, 20, 21, 30$B: $12, 13, 20, 27, 28$

  Road to 800

  Show the answer

  Answer

  Set B

  Type in Desmos

  $A=\left[10,19,20,21,30\right]$$B=\left[12,13,20,27,28\right]$$\operatorname{stdev}\left(A\right)$$\operatorname{stdev}\left(B\right)$

  Where to look

  The bigger number means more spread: B’s $7.52$ beats A’s $7.11$.

  

  Graph made with Desmos

  Tip Range can fool you: A’s is bigger.

• Data and probability

  Remove the $50$. Which changes more, the mean or the median?

  $4, 6, 7, 9, 50$

  Road to 800

  Show the answer

  Answer

  The mean: $15.2$ to $6.5$

  Type in Desmos

  $L=\left[4,6,7,9,50\right]$$M=\left[4,6,7,9\right]$$\left[\operatorname{mean}\left(L\right),\operatorname{median}\left(L\right)\right]$$\left[\operatorname{mean}\left(M\right),\operatorname{median}\left(M\right)\right]$

  Where to look

  Mean drops $15.2$ to $6.5$. Median only drops $7$ to $6.5$.

  

  Graph made with Desmos

• Data and probability

  Find the missing test score.

  Scores: $78, 85, 91$, ?Mean of all four: $86$

  Road to 800

  Show the answer

  Answer

  $90$

  Type in Desmos

  $y=\operatorname{mean}\left(\left[78,85,91,x\right]\right)$$y=86$

  Where to look

  Click where the two lines cross: $(90, 86)$.

  

  Graph made with Desmos

• Data and probability

  Find the line of best fit.

  $x$: $2, 4, 6, 8$$y$: $5, 9, 12, 16$

  Road to 800

  Show the answer

  Answer

  $y = 1.8x + 1.5$

  Type in Desmos

  $x_{1}=\left[2,4,6,8\right]$$y_{1}=\left[5,9,12,16\right]$$y_{1}\sim mx_{1}+b$

  Where to look

  Read $m$ and $b$ under the last row: $m = 1.8$, $b = 1.5$.

  

  Graph made with Desmos

  Tip Use ~ (not =) so Desmos finds the best $m$ and $b$.

• Data and probability

  Find the residual for the data point $(7, 13)$.

  Best-fit line: $y = 1.6x + 3$

  Road to 800

  Show the answer

  Answer

  $-1.2$

  Type in Desmos

  $f\left(x\right)=1.6x+3$$13-f\left(7\right)$

  Where to look

  Read $-1.2$ on the right: the point is below the line.

  

  Graph made with Desmos

  Tip Residual = actual $y$ − predicted $y$.

• Data and probability

  Use the line of best fit to predict $y$ when $x = 12$.

  $x$: $1, 3, 5, 7$$y$: $4, 9, 13, 18$

  Road to 800

  Show the answer

  Answer

  About $29.4$

  Type in Desmos

  $x_{1}=\left[1,3,5,7\right]$$y_{1}=\left[4,9,13,18\right]$$y_{1}\sim mx_{1}+b$$12m+b$

  Where to look

  Read the last row: $29.4$.

  

  Graph made with Desmos

  Tip After the fit, $m$ and $b$ are numbers you can use.

• Data and probability

  Two tiles are drawn without replacement. What is the probability that both are blue?

  Bag: $7$ blue tiles, $5$ white tiles

  Road to 800

  Show the answer

  Answer

  $\dfrac{7}{22}$

  Type in Desmos

  $\dfrac{7}{12}\cdot\dfrac{6}{11}$Keys: 7/12 → *6/11

  Where to look

  Click the fraction button at the left of the row: $\dfrac{7}{22}$.

  

  Graph made with Desmos

  Tip Second draw: $6$ blue left, $11$ tiles left.

• Data and probability

  Which grade has the highest probability that a student picked at random from it bikes to school?

  Grade 9: $27$ bike, $63$ don’tGrade 10: $30$ bike, $70$ don’tGrade 11: $22$ bike, $33$ don’t

  Road to 800

  Show the answer

  Answer

  Grade 11 ($0.4$)

  Type in Desmos

  $B=\left[27,30,22\right]$$N=\left[63,70,33\right]$$\dfrac{B}{\left(B+N\right)}$

  Where to look

  Row 3 gives each grade’s probability, in order: $0.3, 0.3, 0.4$.

  

  Graph made with Desmos

  Tip $B + N$ is each grade’s total.

• Data and probability

  How many green marbles must be added so that the probability of drawing green is $0.4$?

  Bag: $9$ green, $21$ others

  Road to 800

  Show the answer

  Answer

  $5$ green marbles

  Type in Desmos

  $9+x=0.4\left(30+x\right)$

  Where to look

  Click where the vertical line meets the $x$-axis: $(5, 0)$.

  

  Graph made with Desmos

  Tip Green, $9 + x$, is $0.4$ of the new total, $30 + x$.

• Data and probability

  Estimate how many of the town’s $2{,}450$ voters support the plan. Give the plausible range too.

  Sample: $57$ of $190$ support itMargin of error: $6$ points

  Road to 800

  Show the answer

  Answer

  About $735$; plausibly $588$ to $882$

  Type in Desmos

  $p=\dfrac{57}{190}$$2450\left(p+\left[-0.06,0,0.06\right]\right)$

  Where to look

  Row 2 shows the low end, the estimate and the high end.

  

  Graph made with Desmos

  Tip $6$ percentage points is $0.06$.

• Data and probability

  Estimate how many of the school’s $1{,}200$ students play a sport.

  Random sample: $28$ of $80$ play a sport

  Road to 800

  Show the answer

  Answer

  About $420$ students

  Type in Desmos

  $\dfrac{28}{80}\cdot1200$Keys: 28/80 → *1200

  Where to look

  Read the answer on the right: $420$.

  

  Graph made with Desmos

  Tip Sample fraction $\times$ the whole school.

• Geometry

  A cylinder has height $5$ and volume $245\pi$. Find its radius.

  $V = \pi r^2 h$

  Road to 800

  Show the answer

  Answer

  Radius $7$

  Type in Desmos

  $5\pi\cdot x^{2}=245\pi$

  Where to look

  Click where the right-hand vertical line meets the $x$-axis: $(7, 0)$.

  

  Graph made with Desmos

  Tip Type $x$ for the radius. Desmos solves for $x$.

• Geometry

  Two cans have the same shape. How many times as tall is the big can?

  volumes: $64$ cm³ and $343$ cm³

  Road to 800

  Show the answer

  Answer

  $1.75$ times as tall

  Type in Desmos

  $\sqrt[3]{\dfrac{343}{64}}$

  Where to look

  Read the number on the right: $1.75$.

  

  Graph made with Desmos

  Tip Volumes: cube root. Areas: square root.

• Geometry

  A cone has radius $6$ and volume $96\pi$. Find its height.

  $V = \dfrac{1}{3}\pi r^2 h$

  Road to 800

  Show the answer

  Answer

  Height $8$

  Type in Desmos

  $\pi\cdot6^{2}\cdot\dfrac{x}{3}=96\pi$Keys: pi*6^2 → *x/3 → =96pi

  Where to look

  Click where the vertical line meets the $x$-axis: $(8, 0)$.

  

  Graph made with Desmos

  Tip Type $x$ for the height. → is the right-arrow key.

• Geometry

  A sphere has radius $3$. Find its volume, to the nearest whole number.

  $V = \dfrac{4}{3}\pi r^3$

  Road to 800

  Show the answer

  Answer

  About $113$

  Type in Desmos

  $\dfrac{4}{3}\cdot\pi\cdot3^{3}$Keys: 4/3 → *pi*3^3

  Where to look

  Read the answer on the right: $113.097$…

  

  Graph made with Desmos

  Tip → is the right-arrow key. It gets you out of the fraction.

• Geometry

  A rectangle has area $84$. Its length is $5$ more than its width. Find the width.

  width $\times$ (width $+\ 5$) $= 84$

  Road to 800

  Show the answer

  Answer

  Width $7$

  Type in Desmos

  $x\left(x+5\right)=84$

  Where to look

  Click where the right-hand vertical line meets the $x$-axis: $(7, 0)$.

  

  Graph made with Desmos

  Tip A width cannot be negative, so ignore the line at $-12$.

• Geometry

  Find the length of $AB$.

  $A(-1, 2)$ and $B(7, 8)$

  Road to 800

  Show the answer

  Answer

  $AB = 10$

  Type in Desmos

  $A=\left(-1,2\right)$$B=\left(7,8\right)$$\operatorname{distance}\left(A,B\right)$

  Where to look

  Read the last row: $10$.

  

  Graph made with Desmos

• Geometry

  Two angles on a straight line are shown. Find $x$.

  $(3x + 20)^\circ$ and $(2x + 10)^\circ$

  Road to 800

  Show the answer

  Answer

  $x = 30$

  Type in Desmos

  $3x+20+2x+10=180$

  Where to look

  Click where the vertical line meets the $x$-axis: $(30, 0)$.

  

  Graph made with Desmos

  Tip Angles on a straight line add to $180^\circ$.

• Geometry

  Triangles $ABC$ and $DEF$ are similar. Find $EF$.

  $AB = 6$, $BC = 15$$DE = 10$

  Road to 800

  Show the answer

  Answer

  $EF = 25$

  Type in Desmos

  $\dfrac{10}{6}=\dfrac{x}{15}$Keys: 10/6 → =x/15

  Where to look

  Click where the vertical line meets the $x$-axis: $(25, 0)$.

  

  Graph made with Desmos

  Tip Big triangle over small triangle, on both sides.

• Geometry

  Find the midpoint of $AB$.

  $A(-3, 4)$ and $B(5, -2)$

  Road to 800

  Show the answer

  Answer

  $(1, 1)$

  Type in Desmos

  $A=\left(-3,4\right)$$B=\left(5,-2\right)$$\operatorname{midpoint}\left(A,B\right)$

  Where to look

  Read the last row: $(1, 1)$.

  

  Graph made with Desmos

  Tip A midpoint averages the $x$-values and the $y$-values.

• Geometry

  Each interior angle of a regular polygon is $150^\circ$. How many sides does it have?

  angle sum $= 180(n - 2)$

  Road to 800

  Show the answer

  Answer

  $12$ sides

  Type in Desmos

  $180\left(x-2\right)=150x$

  Where to look

  Click where the vertical line meets the $x$-axis: $(12, 0)$.

  

  Graph made with Desmos

  Tip The angles add to $180(n - 2)$, and also to $150n$.

• Geometry

  Right triangle. Find $x$.

  legs: $x$ and $x + 7$hypotenuse: $17$

  Road to 800

  Show the answer

  Answer

  $x = 8$

  Type in Desmos

  $x^{2}+\left(x+7\right)^{2}=17^{2}$

  Where to look

  Click where the right-hand vertical line meets the $x$-axis: $(8, 0)$.

  

  Graph made with Desmos

  Tip The hypotenuse (longest side) goes alone after the =.

• Geometry

  The hypotenuse is $12$ and one leg is $8$. Which is the other leg?

  A) $4\sqrt{5}$ B) $2\sqrt{13}$C) $4\sqrt{13}$ D) $8\sqrt{2}$

  Road to 800

  Show the answer

  Answer

  A) $4\sqrt{5}$

  Type in Desmos

  $\sqrt{144-64}$$4\sqrt{5}$

  Where to look

  Both rows show $8.944$…, so choice A matches.

  

  Graph made with Desmos

  Tip For a leg, subtract: $12^2 - 8^2 = 144 - 64$.

• Geometry

  A $45^\circ$-$45^\circ$-$90^\circ$ triangle has a hypotenuse of $10$. Which is the length of a leg?

  A) $5$ B) $5\sqrt{2}$C) $5\sqrt{3}$ D) $10\sqrt{2}$

  Road to 800

  Show the answer

  Answer

  B) $5\sqrt{2}$

  Type in Desmos

  $\dfrac{10}{\sqrt{2}}$$5\sqrt{2}$

  Where to look

  Both rows show $7.071$…, so choice B matches.

  

  Graph made with Desmos

  Tip A leg is the hypotenuse divided by $\sqrt{2}$.

• Geometry

  Find the center and radius.

  $x^2 + y^2 - 8x + 2y - 8 = 0$

  Road to 800

  Show the answer

  Answer

  Center $(4, -1)$, radius $5$

  Type in Desmos

  $x^{2}+y^{2}-8x+2y-8=0$

  Where to look

  Click the top dot and the bottom dot: $(4, 4)$ and $(4, -6)$.

  

  Graph made with Desmos

  Tip The center is halfway between them.

• Geometry

  Is the point $(5, 1)$ inside, on, or outside the circle?

  $(x - 2)^2 + (y + 1)^2 = 20$

  Road to 800

  Show the answer

  Answer

  Inside

  Type in Desmos

  $\left(x-2\right)^{2}+\left(y+1\right)^{2}<20$$\left(5,1\right)$

  Where to look

  The dot sits in the shaded part, so it’s inside.

  

  Graph made with Desmos

  Tip Too close to tell? Plug the point in.

• Geometry

  Where does the line cross the circle?

  $x^2 + y^2 = 25$$y = x + 1$

  Road to 800

  Show the answer

  Answer

  $(3, 4)$ and $(-4, -3)$

  Type in Desmos

  $x^{2}+y^{2}=25$$y=x+1$

  Where to look

  Click the two points where they cross.

  

  Graph made with Desmos

• Geometry

  The line $y = k$ touches this circle at just one point, and $k > 0$. Find $k$.

  $x^2 + y^2 - 6x + 8y = 0$

  Road to 800

  Show the answer

  Answer

  $k = 1$

  Type in Desmos

  $x^{2}+y^{2}-6x+8y=0$

  Where to look

  Click the top dot: $(3, 1)$. The line $y = 1$ touches there.

  

  Graph made with Desmos

• Geometry

  Find the arc length.

  radius $12$, central angle $105^\circ$

  Road to 800

  Show the answer

  Answer

  $7\pi$ (about $21.99$)

  Type in Desmos

  $2\cdot12\cdot\dfrac{105}{360}$

  Where to look

  It shows $7$, so the arc is $7\pi$.

  

  Graph made with Desmos

  Tip Leave $\pi$ out, then put it back at the end.

• Trigonometry

  Find the hypotenuse, to the nearest tenth.

  $35^\circ$ angle, opposite side $= 14$

  Road to 800

  Show the answer

  Answer

  About $24.4$

  Type in Desmos

  $x\cdot\sin\left(35\right)=14$

  Where to look

  Click where the vertical line meets the $x$-axis: $(24.40826, 0)$.

  

  Graph made with Desmos

  Tip Set Degrees first: click the wrench, then Degrees.

• Trigonometry

  Angle $A$ is acute. Find $\tan A$.

  $\cos A = \dfrac{5}{13}$

  Road to 800

  Show the answer

  Answer

  $\tan A = \dfrac{12}{5}$

  Type in Desmos

  $\tan\left(\arccos\left(\dfrac{5}{13}\right)\right)$

  Where to look

  It shows $2.4$, which is $\dfrac{12}{5}$.

  

  Graph made with Desmos

  Tip Works in Degrees or Radians mode.

• Trigonometry

  Find the exact value.

  $\cos\left(\dfrac{4\pi}{3}\right)$

  Road to 800

  Show the answer

  Answer

  $-\dfrac{1}{2}$

  Type in Desmos

  $\cos\left(\dfrac{4\pi}{3}\right)$

  Where to look

  Read the value: $-0.5$.

  

  Graph made with Desmos

  Tip Needs Radians mode (the default), not Degrees.