Lines and linear functions

Chapter 2 · 7 lessons · 296 questions

Unit 1: Slope and y-intercept

Key points

Every line has an equation: $y = mx + b$ .
$m$ is the slope: rise over run. That is how far the line goes up, divided by how far it goes across.
A line that goes up from left to right has a positive slope. A line that goes down has a negative slope.
$b$ is the $y$ -coordinate of the $y$ -intercept: the line crosses the $y$ -axis at $(0, b)$ .
Get $y$ alone first. $2y = 4x + 6$ becomes $y = 2x + 3$ , so the slope is $2$ .

See it solved

Example

A line passes through the points $(2, 1)$ and $(6, 9)$ . Which equation defines the line?

A $y = 2x - 3$
B $y = 2x + 1$
C $y = \dfrac{1}{2}x$
D $y = 4x - 7$

Worked by hand

The plan

Determine the slope
Determine the $y$ -coordinate of the $y$ -intercept

Desmos: With two points, by hand takes two short steps. Desmos gives the same $m$ and $b$ , so use it as a quick check.

Step 1: Determine the slope
- The rise: $9 - 1 = 8$ .
- The run: $6 - 2 = 4$ .
- $m = \dfrac{8}{4} = 2$ .
- That rules out choices C and D.
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the point $(2, 1)$ into $y = 2x + b$ .
- $1 = 2 \cdot 2 + b$ .
- $2 \cdot 2 = 4$ , so $1 = 4 + b$ .
- $b = 1 - 4 = -3$ .
- The equation is $y = 2x - 3$ . That rules out choice B.

The correct answer choice is A:

y = 2x - 3

The same example in Desmos

new graph, then Add item + and Table
Table
x₁ y₁
2 1
6 9
Desmos shows: The two points.
Keys: y1~mx1+b
Desmos shows: $m = 2$ and $b = -3$ . That is $y = 2x - 3$ : choice A.

Table
x₁	y₁
2	1
6	9

Desmos expression list: a table of x1: 2, 6 and y1: 1, 9; y1~ mx1+b — Type

Desmos graph of a table of x1: 2, 6 and y1: 1, 9; y1~ mx1+b, with (0, −3) labelled — Look

Made with Desmos

Open Desmos

Your turn

A line has a slope of $-3$ and passes through the point $(4, -2)$ . What is the $y$ -coordinate of its $y$ -intercept?

Show the answerHide the answer

Answer

The correct answer is $10$ .

Step 1: Determine the slope
- The slope is given: $m = -3$ .
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug $x = 4$ , $y = -2$ and $m = -3$ into $y = mx + b$ .
- $-2 = -3 \cdot 4 + b$ .
- $-3 \cdot 4 = -12$ , so $-2 = -12 + b$ .
- $b = -2 + 12 = 10$ .

Traps

Slope is rise over run, not run over rise. $4 \div 8 = \dfrac{1}{2}$ . That is choice C, a wrong answer.
Get $y$ alone before you read the slope. In $2y = 4x + 6$ , the slope is not $4$ . Divide by $2$ first: $y = 2x + 3$ . The slope is $2$ .

Practice this lesson24 questions

Unit 2: Standard form

Key points

Standard form is $Ax + By = C$ .
The slope is $-\dfrac{A}{B}$ .
The $y$ -intercept is $\left(0, \dfrac{C}{B}\right)$ . You get it by setting $x = 0$ .
The $x$ -intercept is $\left(\dfrac{C}{A}, 0\right)$ . You get it by setting $y = 0$ .

See it solved

Example

In the $xy$ -plane, what are the slope and the $y$ -intercept of the graph of $3x + 4y = 24$ ?

ASlope $-\dfrac{3}{4}$ , $y$ -intercept $(0, 6)$
BSlope $-\dfrac{3}{4}$ , $y$ -intercept $(0, 8)$
CSlope $\dfrac{3}{4}$ , $y$ -intercept $(0, 6)$
DSlope $-\dfrac{4}{3}$ , $y$ -intercept $(0, 6)$

Worked by hand

The plan

Determine the slope
Determine the $y$ -coordinate of the $y$ -intercept

Desmos: Reading $-\dfrac{A}{B}$ and $\dfrac{C}{B}$ takes two lines, so by hand is quicker. In Desmos, type the equation as given and click the gray points to check.

Step 1: Determine the slope
- Match $3x + 4y = 24$ with $Ax + By = C$ : $A = 3$ , $B = 4$ and $C = 24$ .
- $m = -\dfrac{A}{B} = -\dfrac{3}{4}$ .
- That rules out choices C and D.
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- $b = \dfrac{C}{B} = \dfrac{24}{4} = 6$ .
- The $y$ -intercept is $(0, 6)$ . That rules out choice B.
- Check: plug $x = 0$ into $3x + 4y = 24$ . $3 \cdot 0 = 0$ , so $4y = 24$ .
- $y = \dfrac{24}{4} = 6$ . It matches.

The correct answer choice is A: the slope is

-\dfrac{3}{4}

and the

y

-intercept is

(0, 6)

The same example in Desmos

new graph
Desmos shows: The line, typed as given.
click the line, then its gray points on the axes
Desmos shows: $(0, 6)$ and $(8, 0)$ . The $y$ -intercept is $(0, 6)$ .
read the slope from the two points
Desmos shows: From $(0, 6)$ to $(8, 0)$ the line goes down $6$ and across $8$ : $-\dfrac{6}{8} = -\dfrac{3}{4}$ .

Desmos graph of 3x+4y=24, with (0, 6) and (8, 0) labelled — Look

Made with Desmos

Open Desmos

Your turn

The graph of $5x - 2y = 20$ crosses the $x$ -axis at the point $(k, 0)$ . What is the value of $k$ ?

Show the answerHide the answer

Answer

The correct answer is $4$ .

Step 1: Determine the $x$ -coordinate of the $x$ -intercept
- On the $x$ -axis, $y = 0$ . Plug $y = 0$ into $5x - 2y = 20$ .
- $5x - 2 \cdot 0 = 20$ .
- $2 \cdot 0 = 0$ , so $5x = 20$ .
- $x = \dfrac{20}{5} = 4$ . So $k = 4$ .

Traps

Forgetting the minus sign. The slope is $-\dfrac{A}{B}$ , not $\dfrac{A}{B}$ . A slope of $\dfrac{3}{4}$ is choice C, a wrong answer.
Flipping the fraction. $A$ , the number with $x$ , goes on top. $-\dfrac{B}{A} = -\dfrac{4}{3}$ is choice D, a wrong answer.

Practice this lesson24 questions

Unit 3: Missing coordinates on a line

Key points

Any two points on a line give the same slope.
Slope $= \dfrac{y_2 - y_1}{x_2 - x_1}$ . Subtract in the same order on the top and on the bottom.
If the slope is known, write one slope equation and solve it.
If the slope is not known, write the slope twice, from two pairs of points, and set the two equal.

See it solved

Example

The points $(1, 4)$ , $(5, 12)$ and $(8, k)$ lie on the same line. What is the value of $k$ ?

Worked by hand

The plan

Set up two slope equations
Equate the two equations

Desmos: Desmos does not find a missing coordinate for you. Use it only as a check: fit a line to the two known points, then read the third.

Step 1: Set up two slope equations
- From $(1, 4)$ and $(5, 12)$ : the slope is $\dfrac{12 - 4}{5 - 1}$ .
- $12 - 4 = 8$ and $5 - 1 = 4$ . So the slope is $\dfrac{8}{4} = 2$ .
- From $(5, 12)$ and $(8, k)$ : the slope is $\dfrac{k - 12}{8 - 5}$ .
- $8 - 5 = 3$ . So the slope is $\dfrac{k - 12}{3}$ .
Step 2: Equate the two equations
- Both are the slope of the same line: $\dfrac{k - 12}{3} = 2$ .
- Multiply both sides by $3$ : $k - 12$ $= 2 \cdot 3$ $= 6$ .
- Add $12$ to both sides: $k = 6 + 12 = 18$ .
- Check: from $(1, 4)$ to $(8, 18)$ the slope is $\dfrac{18 - 4}{8 - 1} = \dfrac{14}{7} = 2$ . It matches.

The correct answer is

18

The same example in Desmos

new graph, then Add item + and Table
Table
x₁ y₁
1 4
5 12
Desmos shows: The two known points.
Keys: y1~mx1+b
Desmos shows: $m = 2$ and $b = 2$ .
Desmos shows: The line through both points.
Desmos shows: $18$ , at the right of the row. So $k = 18$ .

Table
x₁	y₁
1	4
5	12

Desmos expression list: a table of x1: 1, 5 and y1: 4, 12; y1~ mx1+b; f(x)=mx+b; f(8) — Type

Desmos graph of a table of x1: 1, 5 and y1: 4, 12; y1~ mx1+b; f(x)=mx+b; f(8), with (8, 18) labelled — Look

Made with Desmos

Open Desmos

Your turn

A line has a slope of $\dfrac{3}{2}$ and passes through the points $(2, 5)$ and $(6, k)$ . What is the value of $k$ ?

Show the answerHide the answer

Answer

The correct answer is $11$ .

Step 1: Set up the slope equation
- The slope from $(2, 5)$ to $(6, k)$ is $\dfrac{k - 5}{6 - 2}$ .
- $6 - 2 = 4$ . So $\dfrac{k - 5}{4} = \dfrac{3}{2}$ .
Step 2: Solve the equation
- Multiply both sides by $4$ : $k - 5 = \dfrac{3}{2} \cdot 4$ .
- $\dfrac{3}{2} \cdot 4 = \dfrac{12}{2} = 6$ . So $k - 5 = 6$ .
- Add $5$ to both sides: $k = 6 + 5 = 11$ .

Traps

Subtracting in a different order on the top and on the bottom. $\dfrac{12 - 4}{1 - 5} = \dfrac{8}{-4} = -2$ is the wrong slope. It gives $k - 12$ $= -2 \cdot 3$ $= -6$ , so $k = 6$ , a wrong answer.
Putting the $x$ -values on top. Slope is the change in $y$ over the change in $x$ . $\dfrac{5 - 1}{12 - 4} = \dfrac{4}{8} = \dfrac{1}{2}$ is run over rise. It gives $k - 12 = \dfrac{3}{2}$ , so $k = 13.5$ , a wrong answer.

Practice this lesson16 questions

Unit 4: Parallel lines

Key points

Parallel lines have the same slope.
They have different $y$ -intercepts. The same slope and the same $y$ -intercept is the same line.
Parallel lines never cross.
To read the slope of the given line, get $y$ alone. For $Ax + By = C$ , use $-\dfrac{A}{B}$ .

See it solved

Example

Line $p$ is parallel to the line $y = 3x - 5$ and passes through the point $(2, 10)$ . Which equation defines line $p$ ?

A $y = -\dfrac{1}{3}x + 4$
B $y = 3x - 5$
C $y = 3x + 4$
D $y = 3x + 10$

Worked by hand

The plan

Determine the slope
Determine the $y$ -coordinate of the $y$ -intercept

Desmos: By hand takes two short steps. In Desmos, a table with the one point and a fit with the slope typed in gives $b$ , so use it as a check.

Step 1: Determine the slope
- The given line is $y = 3x - 5$ . Its slope is $3$ .
- Parallel lines have the same slope, so line $p$ has slope $3$ .
- That rules out choice A.
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the point $(2, 10)$ into $y = 3x + b$ .
- $10 = 3 \cdot 2 + b$ .
- $3 \cdot 2 = 6$ , so $10 = 6 + b$ .
- $b = 10 - 6 = 4$ .
- Line $p$ is $y = 3x + 4$ . That rules out choices B and D.

The correct answer choice is C:

y = 3x + 4

The same example in Desmos

new graph
Desmos shows: The given line.
Add item + and Table
Table
x₁ y₁
2 10
Desmos shows: The point $(2, 10)$ .
Keys: y1~3x1+b
Desmos shows: $b = 4$ . Only $b$ is found, because the slope $3$ is typed in.
Desmos shows: A second line, through the point and parallel to the first: choice C.

Table
x₁	y₁
2	10

Desmos expression list: y=3x-5; a table of x1: 2 and y1: 10; y1~3x1+b; y=3x+4 — Type

Desmos graph of y=3x-5; a table of x1: 2 and y1: 10; y1~3x1+b; y=3x+4, with (0, 4) labelled — Look

Made with Desmos

Open Desmos

Your turn

Line $t$ is parallel to the line $2x + 5y = 10$ and passes through the point $(5, 1)$ . What is the $y$ -coordinate of the $y$ -intercept of line $t$ ?

Show the answerHide the answer

Answer

The correct answer is $3$ .

Step 1: Determine the slope
- For $2x + 5y = 10$ , $A = 2$ and $B = 5$ .
- The slope is $-\dfrac{A}{B} = -\dfrac{2}{5}$ .
- Line $t$ is parallel, so its slope is also $-\dfrac{2}{5}$ .
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the point $(5, 1)$ into $y = -\dfrac{2}{5}x + b$ .
- $1 = -\dfrac{2}{5} \cdot 5 + b$ .
- $-\dfrac{2}{5} \cdot 5$ $= -\dfrac{10}{5}$ $= -2$ , so $1 = -2 + b$ .
- $b = 1 + 2 = 3$ .

Traps

Using the negative reciprocal. $-\dfrac{1}{3}$ is the slope of a perpendicular line, not a parallel one. That is choice A, a wrong answer.
Using the point's $y$ -coordinate as $b$ . The point $(2, 10)$ is not on the $y$ -axis, so $10$ is not $b$ . $y = 3x + 10$ is choice D, a wrong answer.

Practice this lesson20 questions

Unit 5: Perpendicular lines

Key points

Perpendicular lines cross at a right angle.
Their slopes are negative reciprocals: flip the fraction, then change the sign.
$\dfrac{2}{3}$ becomes $-\dfrac{3}{2}$ . $-4$ becomes $\dfrac{1}{4}$ .
Multiply the two slopes and you always get $-1$ .
An $x$ -intercept is a point too: $(x, 0)$ .

See it solved

Example

Line $r$ is perpendicular to the line $y = \dfrac{2}{3}x + 1$ and passes through the point $(4, -1)$ . Which equation defines line $r$ ?

A $y = -\dfrac{3}{2}x + 5$
B $y = -\dfrac{3}{2}x - 1$
C $y = \dfrac{2}{3}x - \dfrac{11}{3}$
D $y = \dfrac{3}{2}x - 7$

Worked by hand

The plan

Determine the slope
Determine the $y$ -coordinate of the $y$ -intercept

Desmos: By hand takes two short steps. In Desmos, a table with the one point and a fit with the new slope typed in gives $b$ , so use it as a check.

Step 1: Determine the slope
- The given line has slope $\dfrac{2}{3}$ .
- Flip it: $\dfrac{3}{2}$ . Change the sign: $-\dfrac{3}{2}$ .
- Check: $\dfrac{2}{3} \cdot \left(-\dfrac{3}{2}\right)$ $= -\dfrac{6}{6}$ $= -1$ .
- Line $r$ has slope $-\dfrac{3}{2}$ . That rules out choices C and D.
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the point $(4, -1)$ into $y = -\dfrac{3}{2}x + b$ .
- $-1 = -\dfrac{3}{2} \cdot 4 + b$ .
- $-\dfrac{3}{2} \cdot 4$ $= -\dfrac{12}{2}$ $= -6$ , so $-1 = -6 + b$ .
- $b = -1 + 6 = 5$ .
- Line $r$ is $y = -\dfrac{3}{2}x + 5$ . That rules out choice B.

The correct answer choice is A:

y = -\dfrac{3}{2}x + 5

The same example in Desmos

new graph
Keys: y=2/3 → x+1
Desmos shows: The given line.
Add item + and Table
Table
x₁ y₁
4 -1
Desmos shows: The point $(4, -1)$ .
Keys: y1~-3/2 → x1+b
Desmos shows: $b = 5$ . Only $b$ is found, because the slope $-\dfrac{3}{2}$ is typed in.
Keys: y=-3/2 → x+5
Desmos shows: A second line, through the point. It crosses the first at a right angle: choice A.

Table
x₁	y₁
4	-1

Desmos expression list: y=frac{2}{3}x+1; a table of x1: 4 and y1: -1; y1~-frac{3}{2}x1+b; y=-frac{3}{2}x+5 — Type

Desmos graph of y=frac{2}{3}x+1; a table of x1: 4 and y1: -1; y1~-frac{3}{2}x1+b; y=-frac{3}{2}x+5, with (0, 5) labelled — Look

Made with Desmos

Open Desmos

Your turn

Line $n$ is perpendicular to the line $y = -4x + 3$ and crosses the $x$ -axis at the point $(8, 0)$ . What is the $y$ -coordinate of the $y$ -intercept of line $n$ ?

Show the answerHide the answer

Answer

The correct answer is $-2$ .

Step 1: Determine the slope
- The given line has slope $-4$ . As a fraction, that is $-\dfrac{4}{1}$ .
- Flip it: $-\dfrac{1}{4}$ . Change the sign: $\dfrac{1}{4}$ .
- Line $n$ has slope $\dfrac{1}{4}$ .
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the point $(8, 0)$ into $y = \dfrac{1}{4}x + b$ .
- $0 = \dfrac{1}{4} \cdot 8 + b$ .
- $\dfrac{1}{4} \cdot 8 = \dfrac{8}{4} = 2$ , so $0 = 2 + b$ .
- $b = 0 - 2 = -2$ .

Traps

Flipping without changing the sign. $\dfrac{3}{2}$ is the reciprocal, but it is not negative. That slope gives choice D, a wrong answer.
Keeping the same slope. $\dfrac{2}{3}$ is the slope of a parallel line. That gives choice C, a wrong answer.

Practice this lesson44 questions

Unit 6: Notation, tables and rules

Key points

$f(x)$ is not $f$ times $x$ . It is the output when the input is $x$ .
$f(3)$ means: put $3$ in, and see what comes out.
In a linear function, each time the input goes up by $1$ , the output changes by the same amount. That amount is the slope, $m$ .
The output when the input is $0$ is $b$ , the $y$ -coordinate of the $y$ -intercept $(0, b)$ . So $f(x) = mx + b$ .
In Desmos, a table plus the tilde key, ~, gives you $m$ and $b$ .

See it solved

Example

A linear function $f$ has $f(1) = 5$ , $f(3) = 11$ and $f(7) = 23$ . What is $f(10)$ ?

Worked by hand

The plan

Determine the slope
Determine the $y$ -coordinate of the $y$ -intercept
Plug the input into the function

Desmos: With three rows, by hand is just as quick. Desmos gives the same rule, so use it as a check, or when the numbers are messy.

Step 1: Determine the slope
- Use the first two rows: $(1, 5)$ and $(3, 11)$ .
- The input goes up by $3 - 1 = 2$ .
- The output goes up by $11 - 5 = 6$ .
- $m = \dfrac{6}{2} = 3$ .
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug the row $(1, 5)$ and $m = 3$ into $y = mx + b$ .
- $5 = 3 \cdot 1 + b$ .
- $3 \cdot 1 = 3$ , so $5 = 3 + b$ .
- $b = 5 - 3 = 2$ .
- The rule is $f(x) = 3x + 2$ .
- Check: the third row gives $3 \cdot 7 + 2$ $= 21 + 2$ $= 23$ . It matches.
Step 3: Plug the input into the function
- $f(10)$ $= 3 \cdot 10 + 2$ .
- $3 \cdot 10 = 30$ .
- $30 + 2 = 32$ .

The correct answer is

32

The same example in Desmos

new graph, then Add item + and Table
Table
x₁ y₁
1 5
3 11
7 23
Desmos shows: Three points.
Keys: y1~mx1+b
Desmos shows: $m = 3$ and $b = 2$ .
Desmos shows: The line goes through all three points.
Desmos shows: $32$ , at the right of the row.

Table
x₁	y₁
1	5
3	11
7	23

Desmos expression list: a table of x1: 1, 3, 7 and y1: 5, 11, 23; y1~ mx1+b; f(x)=mx+b; f(10) — Type

Desmos graph of a table of x1: 1, 3, 7 and y1: 5, 11, 23; y1~ mx1+b; f(x)=mx+b; f(10), with (10, 32) labelled — Look

Made with Desmos

Open Desmos

Your turn

A linear function $g$ has $g(2) = 7$ and $g(6) = -5$ . What is $g(0)$ ?

Show the answerHide the answer

Answer

The correct answer is $13$ .

Step 1: Determine the slope
- The input goes up by $6 - 2 = 4$ .
- The output changes by $-5 - 7 = -12$ . It goes down by $12$ .
- $m = \dfrac{-12}{4} = -3$ .
Step 2: Determine the $y$ -coordinate of the $y$ -intercept
- Plug $(2, 7)$ and $m = -3$ into $y = mx + b$ .
- $7 = -3 \cdot 2 + b$ .
- $-3 \cdot 2 = -6$ , so $7 = -6 + b$ .
- $b = 7 + 6 = 13$ .
- $g(0)$ is $b$ . So $g(0) = 13$ .

Traps

The first number in a table is not always $b$ . $b$ is the output when $x = 0$ . Here the first input is $1$ , so $5$ is not $b$ . Using $5$ as $b$ gives $3 \cdot 10 + 5$ $= 30 + 5$ $= 35$ , a wrong answer.
The inputs do not go up by $1$ . From $x = 1$ to $x = 3$ is $3 - 1 = 2$ steps, so the slope is $6 \div 2 = 3$ , not $11 - 5 = 6$ . With $m = 6$ : $5 = 6 \cdot 1 + b$ , so $b = 5 - 6 = -1$ . Then $f(10)$ $= 6 \cdot 10 - 1$ $= 60 - 1$ $= 59$ , a wrong answer.

Practice this lesson88 questions

Unit 7: Shifts and reflections of a line

Key points

$f(x) + k$ moves the graph $k$ units up. The change is outside the parentheses, so it does what it says.
$f(x - h)$ moves the graph $h$ units right.
Inside the parentheses works backwards: minus means right, plus means left.
$-f(x)$ reflects the graph across the $x$ -axis: every term changes sign.
$f(-x)$ reflects the graph across the $y$ -axis: only the $x$ -term changes sign.

See it solved

Example

$f(x) = 3x - 2$ . The graph of $y = g(x)$ is the graph of $y = f(x)$ shifted $2$ units right and $4$ units up. What is $g(x)$ ?

Worked by hand

The plan

Determine the translation
Plug the input into the function

Desmos: Desmos is quicker when there are answer choices: type the shift as the question says it, then type each choice and keep the one that lies on top.

Step 1: Determine the translation
- $2$ units right means $x - 2$ , inside the parentheses.
- $4$ units up means $+\,4$ , outside.
- So $g(x)$ $= f(x - 2) + 4$ .
Step 2: Plug the input into the function
- Replace every $x$ in $f$ with $(x - 2)$ .
- $f(x - 2)$ $= 3(x - 2) - 2$ .
- $3 \cdot x = 3x$ and $3 \cdot (-2) = -6$ .
- So $3(x - 2) - 2$ $= 3x - 6 - 2$ .
- $-6 - 2 = -8$ . So $f(x - 2)$ $= 3x - 8$ .
- Now add $4$ : $g(x)$ $= 3x - 8 + 4$ .
- $-8 + 4 = -4$ . So $g(x) = 3x - 4$ .
- Check: the intercept moved from $-2$ to $-4$ .
- The shift right took away the slope times the shift: $3 \cdot 2 = 6$ .
- The shift up added $4$ : $-2 - 6 = -8$ and $-8 + 4 = -4$ .

The correct answer is

g(x) = 3x - 4

The same example in Desmos

new graph
Desmos shows: The original line.
Desmos shows: A second line, parallel to the first.
click the new line, then its gray point on the y-axis
Desmos shows: $(0, -4)$ : the new $y$ -intercept.
Desmos shows: It lies exactly on top of $g$ (dashed in the picture). So $g(x) = 3x - 4$ .

Desmos expression list: f(x)=3x-2; g(x)=f(x-2)+4; y=3x-4 — Type

Desmos graph of f(x)=3x-2; g(x)=f(x-2)+4; y=3x-4, with (0, −4) labelled — Look

Made with Desmos

Open Desmos

Your turn

$f(x) = 4x - 8$ . The graph of $y = g(x)$ is the graph of $y = f(x)$ reflected across the $x$ -axis, then shifted $2$ units left. What is $g(x)$ ?

Show the answerHide the answer

Answer

The correct answer is $g(x) = -4x$ .

Step 1: Determine the translation
- Reflect first: every term changes sign. $-f(x) = -4x + 8$ .
- $2$ units left means: replace $x$ with $(x + 2)$ .
Step 2: Plug the input into the function
- That gives $-4(x + 2) + 8$ .
- $-4 \cdot x = -4x$ and $-4 \cdot 2 = -8$ .
- So $-4(x + 2) + 8$ $= -4x - 8 + 8$ .
- $-8 + 8 = 0$ . So $g(x) = -4x$ .

Traps

Going the wrong way. $f(x - 2)$ moves the graph $2$ units right, not left. Moving left gives $3(x + 2) - 2 + 4$ $= 3x + 6 - 2 + 4$ . $6 - 2 = 4$ and $4 + 4 = 8$ , so that is $3x + 8$ , a wrong answer.
Not multiplying by the slope. $3(x - 2)$ is $3x - 6$ , not $3x - 2$ . That slip gives $3x - 2 - 2 + 4$ . $-2 - 2 = -4$ and $-4 + 4 = 0$ , so that is $3x$ , a wrong answer.

Practice this lesson80 questions

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
Type y1 and Desmos shows y₁: a digit typed straight after a letter becomes a subscript. If you type the underscore yourself (y_1), press → after the digit: a ~ typed next lands inside the subscript (+, − and = leave it by themselves).
The ~ key is Shift + ` on most keyboards (the key left of 1). Use ~, not =: ~ asks Desmos to find the unknown letters; = only draws or defines.
Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
A letter that already has a value, such as a slider m = 1 left over from earlier, is held fixed by a ~ fit, with no warning. Start a new graph, or delete old sliders, before each question.
Start a new graph for each example. Rows left from an earlier example clash with the new ones: a second $f(x)$ , or a second fit for $m$ and $b$ , is an error.
Type y1 and Desmos shows $y_1$ : a digit typed straight after a letter becomes a subscript.
The tilde, ~, is Shift and the key left of 1 on most keyboards. Use ~, not =: ~ finds the letters that make the line fit; = only draws or defines.
Desmos works in $x$ and $y$ . Type any other letter, such as $h$ or $t$ , as $x$ .

Start practicing

Practice by lesson

Lesson 1, Slope and y-intercept, 24 questions
Lesson 2, Standard form, 24 questions
Lesson 3, Missing coordinates on a line, 16 questions
Lesson 4, Parallel lines, 20 questions
Lesson 5, Perpendicular lines, 44 questions
Lesson 6, Notation, tables and rules, 88 questions
Lesson 7, Shifts and reflections of a line, 80 questions

Mixed practice: every lesson

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Easy set 1

11 questions

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All easy sets

Set 1, 11 questions
Set 2, 11 questions
Set 3, 11 questions
Set 4, 11 questions
Set 5, 10 questions
Set 6, 10 questions

Set 1, 10 questions
Set 2, 10 questions
Set 3, 10 questions
Set 4, 10 questions
Set 5, 10 questions
Set 6, 10 questions
Set 7, 10 questions
Set 8, 10 questions
Set 9, 10 questions
Set 10, 9 questions
Set 11, 9 questions

Set 1, 11 questions
Set 2, 11 questions
Set 3, 11 questions
Set 4, 11 questions
Set 5, 10 questions
Set 6, 10 questions
Set 7, 10 questions
Set 8, 10 questions
Set 9, 10 questions
Set 10, 10 questions
Set 11, 10 questions
Set 12, 10 questions

Next chapter · Chapter 3Systems of equations and inequalities

Desmos

Next set

1Unit 1: Slope and y-intercept

Key points

See it solved

Worked by hand

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

The same example in Desmos

Your turn

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

Traps

2Unit 2: Standard form

Key points

See it solved

Worked by hand

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

The same example in Desmos

Your turn

Step 1: Determine the xxx-coordinate of the xxx-intercept

Traps

3Unit 3: Missing coordinates on a line

Key points

See it solved

Worked by hand

Step 1: Set up two slope equations

Step 2: Equate the two equations

The same example in Desmos

Your turn

Step 1: Set up the slope equation

Step 2: Solve the equation

Traps

4Unit 4: Parallel lines

Key points

See it solved

Worked by hand

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

The same example in Desmos

Your turn

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

Traps

5Unit 5: Perpendicular lines

Key points

See it solved

Worked by hand

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

The same example in Desmos

Your turn

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

Traps

6Unit 6: Notation, tables and rules

Key points

See it solved

Worked by hand

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

Step 3: Plug the input into the function

The same example in Desmos

Your turn

Step 1: Determine the slope

Step 2: Determine the yyy-coordinate of the yyy-intercept

Traps

7Unit 7: Shifts and reflections of a line

Key points

See it solved

Worked by hand

Step 1: Determine the translation

Step 2: Plug the input into the function

The same example in Desmos

Your turn

Step 1: Determine the translation

Step 2: Plug the input into the function

Traps

Practice by lesson

Mixed practice: every lesson

Easy set 1

Unit 1: Slope and y-intercept

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Unit 2: Standard form

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Step 1: Determine the $x$ -coordinate of the $x$ -intercept

Unit 3: Missing coordinates on a line

Unit 4: Parallel lines

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Unit 5: Perpendicular lines

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Unit 6: Notation, tables and rules

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Step 2: Determine the $y$ -coordinate of the $y$ -intercept

Unit 7: Shifts and reflections of a line