Quadratics and parabolas

Chapter 6 · 11 lessons · 472 questions

Unit 1: Solving a quadratic

Key points

First set the equation equal to $0$ .
If $a = 1$ , find two numbers that multiply to $c$ and add to $b$ . If $a$ is not $1$ , use two numbers that multiply to $ac$ and add to $b$ .
Each factor gives a root: set the factor equal to $0$ .
Roots are also called solutions, zeros and $x$ -intercepts.
If the quadratic does not factor, use the quadratic formula: $x$ $= \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

See it solved

Example

$3x^2 + 10x - 8$ $= 0$

What is the positive solution to the given equation?

Worked by hand

The plan

Determine two factors of $ac$ that add to $b$
Split the value of $b$
Factor by grouping
Determine the roots

Desmos: Desmos is quicker here: type the left side as $y = \ldots$ and click the $x$ -intercepts.

Step 1: Determine two factors of $ac$ that add to $b$
- $a = 3$ , $b = 10$ and $c = -8$ .
- $ac$ $= 3 \cdot (-8)$ $= -24$ .
- Two numbers that multiply to $-24$ and add to $10$ : they are $12$ and $-2$ .
- Check: $12 \cdot (-2) = -24$ and $12 + (-2) = 10$ .
Step 2: Split the value of $b$
- Write $10x$ as $12x - 2x$ .
- $3x^2 + 12x - 2x - 8$ $= 0$ .
Step 3: Factor by grouping
- First pair: $3x^2 + 12x$ $= 3x(x + 4)$ .
- Second pair: $-2x - 8$ $= -2(x + 4)$ .
- Both pairs have $(x + 4)$ : $(3x - 2)(x + 4)$ $= 0$ .
Step 4: Determine the roots
- $3x - 2 = 0$ gives $3x = 2$ , so $x = \dfrac{2}{3}$ .
- $x + 4 = 0$ gives $x = -4$ .
- The positive solution is $\dfrac{2}{3}$ .

The correct answer is

\dfrac{2}{3}

The same example in Desmos

new graph
Keys: y=3x^2 → +10x-8
Desmos shows: A parabola. Zoom out to see where it meets the $x$ -axis.
click the points where the curve meets the x-axis
Desmos shows: $(-4, 0)$ and about $(0.667, 0)$ . The positive solution is about $0.667$ , which is $\dfrac{2}{3}$ .

Desmos expression list: y=3x^2+10x-8 — Type

Desmos graph of y=3x^2+10x-8, with (−4, 0) and (0.6666666666666666, 0) labelled — Look

Made with Desmos

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Your turn

$x^2 - 3x = 28$

What is the positive solution to the given equation?

Show the answerHide the answer

Answer

The correct answer is $7$ .

Step 1: Set the equation equal to $0$
- Subtract $28$ from both sides: $x^2 - 3x - 28$ $= 0$ .
Step 2: Determine two factors of $c$ that add to $b$
- Two numbers that multiply to $-28$ and add to $-3$ : they are $-7$ and $4$ .
- $(x - 7)(x + 4)$ $= 0$ .
Step 3: Determine the roots
- $x - 7 = 0$ gives $x = 7$ .
- $x + 4 = 0$ gives $x = -4$ .
- The positive solution is $7$ .

Traps

Factoring before the equation equals $0$ . From $x(x - 3) = 28$ you cannot say $x = 28$ or $x - 3 = 28$ . That only works when the other side is $0$ . Move the $28$ first.
Giving a root the sign of its factor. $(x + 4) = 0$ gives $x = -4$ , not $4$ . And $(3x - 2) = 0$ gives $\dfrac{2}{3}$ , not $2$ .

Practice this lesson56 questions

Unit 2: Hidden quadratics

Key points

Some equations are quadratics in disguise: one piece appears squared and also on its own.
Rename the repeated piece with a new letter, such as $u$ .
Solve the quadratic in $u$ , then put the piece back in place of $u$ .
Reject values that are impossible: $\sqrt{x}$ and $x^2$ cannot be negative.

See it solved

Example

$x^4 - 13x^2 + 36$ $= 0$

What is the greatest solution to the given equation?

Worked by hand

The plan

Rename the repeated piece
Determine the roots
Substitute back

Desmos: Desmos is quicker here: type the equation straight in as $y = \ldots$ and click the $x$ -intercepts.

Step 1: Rename the repeated piece
- $x^4$ is $(x^2)^2$ . Let $u = x^2$ .
- The equation becomes $u^2 - 13u + 36$ $= 0$ .
Step 2: Determine the roots
- Two numbers that multiply to $36$ and add to $-13$ : they are $-4$ and $-9$ .
- $(u - 4)(u - 9)$ $= 0$ , so $u = 4$ or $u = 9$ .
Step 3: Substitute back
- $x^2 = 4$ gives $x = 2$ or $x = -2$ .
- $x^2 = 9$ gives $x = 3$ or $x = -3$ .
- The four solutions are $-3$ , $-2$ , $2$ and $3$ . The greatest is $3$ .
- Check: $3^4 - 13 \cdot 3^2 + 36$ $= 81 - 117 + 36$ $= 0$ .

The correct answer is

3

The same example in Desmos

new graph
Keys: y=x^4 → -13x^2 → +36
Desmos shows: A W-shaped curve.
click the points where the curve meets the x-axis
Desmos shows: $(-3, 0)$ , $(-2, 0)$ , $(2, 0)$ and $(3, 0)$ . The greatest solution is $3$ .

Desmos expression list: y=x^4-13x^2+36 — Type

Desmos graph of y=x^4-13x^2+36, with (−3, 0) and (−2, 0) and (2, 0) and (3, 0) labelled — Look

Made with Desmos

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Your turn

$x - 2\sqrt{x} - 8 = 0$

What is the solution to the given equation?

Show the answerHide the answer

Answer

The correct answer is $16$ .

Step 1: Rename the repeated piece
- $x$ is $(\sqrt{x})^2$ . Let $u = \sqrt{x}$ .
- The equation becomes $u^2 - 2u - 8 = 0$ .
Step 2: Determine the roots
- Two numbers that multiply to $-8$ and add to $-2$ : they are $-4$ and $2$ .
- $(u - 4)(u + 2)$ $= 0$ , so $u = 4$ or $u = -2$ .
Step 3: Substitute back
- $\sqrt{x} = -2$ is impossible, because a square root cannot be negative. Reject it.
- $\sqrt{x} = 4$ gives $x = 4^2 = 16$ .
- Check: $16 - 2 \cdot \sqrt{16} - 8$ $= 16 - 2 \cdot 4 - 8$ $= 16 - 8 - 8$ $= 0$ .

Traps

Stopping at $u$ . $u = 4$ and $u = 9$ are values of $x^2$ , not of $x$ . Answering $9$ is wrong: take the square roots.
Forgetting the negative roots. $x^2 = 9$ gives $3$ and $-3$ . Here the greatest is still $3$ , but a question about how many solutions needs all four.

Practice this lesson32 questions

Unit 3: How many solutions? The discriminant

Key points

Write the equation as $ax^2 + bx + c$ $= 0$ and name $a$ , $b$ and $c$ .
The discriminant is $b^2 - 4ac$ .
If it is positive, there are two real solutions. If it is $0$ , there is exactly one. If it is negative, there are none.
With an unknown constant: for exactly one solution, set the discriminant equal to $0$ . For no solution, make it less than $0$ .

See it solved

Example

$2x^2 - 12x + c$ $= 0$

In the given equation, $c$ is a constant. The equation has exactly one real solution. What is the value of $c$ ?

Worked by hand

The plan

Set the discriminant equal to $0$
Solve for $c$

Desmos: With a constant to find, use the discriminant by hand. Desmos is a check: graph the equation with your value and count where it meets the $x$ -axis.

Step 1: Set the discriminant equal to $0$
- $a = 2$ and $b = -12$ . The constant $c$ is unknown.
- Exactly one solution means $b^2 - 4ac = 0$ .
- $(-12)^2 - 4 \cdot 2 \cdot c$ $= 0$ .
- $(-12)^2 = 144$ and $4 \cdot 2 = 8$ . So $144 - 8c = 0$ .
Step 2: Solve for $c$
- Add $8c$ to both sides: $144 = 8c$ .
- $c = \dfrac{144}{8} = 18$ .
- Check: $2x^2 - 12x + 18$ $= 2(x^2 - 6x + 9)$ $= 2(x - 3)^2$ . It is $0$ only when $x = 3$ .

The correct answer is

18

The same example in Desmos

new graph
Keys: y=2x^2 → -12x+18
Desmos shows: The parabola with $c = 18$ . It touches the $x$ -axis.
click the point where it touches the x-axis
Desmos shows: $(3, 0)$ , and nowhere else. One solution, so $c = 18$ is right.

Desmos expression list: y=2x^2-12x+18 — Type

Desmos graph of y=2x^2-12x+18, with (3, 0) labelled — Look

Made with Desmos

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Your turn

$3x^2 - 5x + 4$ $= 0$

How many distinct real solutions does the given equation have?

AZero
BExactly one
CExactly two
DInfinitely many

Show the answerHide the answer

Answer

The correct answer choice is A: the discriminant is $-23$ , so there are no real solutions.

Step 1: Determine the number of solutions using the discriminant
- $a = 3$ , $b = -5$ and $c = 4$ .
- $b^2 - 4ac$ $= (-5)^2 - 4 \cdot 3 \cdot 4$ .
- $(-5)^2 = 25$ and $4 \cdot 3 \cdot 4 = 48$ .
- $25 - 48 = -23$ .
- The discriminant is negative, so there are no real solutions.

Traps

Squaring $-12$ to a negative number. $(-12)^2 = 144$ , not $-144$ . With $-144$ you get $-144 - 8c = 0$ and $c = -18$ , a wrong answer.
Leaving out the $4$ . $b^2 - ac$ gives $144 - 2c = 0$ and $c = 72$ , a wrong answer. The discriminant is $b^2 - 4ac$ .

Practice this lesson56 questions

Unit 4: Sum and product of the roots

Key points

Write the equation as $ax^2 + bx + c$ $= 0$ first.
The sum of the roots is $-\dfrac{b}{a}$ .
The product of the roots is $\dfrac{c}{a}$ .
You do not need to find the roots themselves.

See it solved

Example

$3x^2 - 12x + 5$ $= 0$

What is the sum of the solutions to the given equation?

Worked by hand

The plan

Determine $a$ , $b$ and $c$
Determine the sum of the roots

Desmos: The formula takes one line. Desmos is a check: click the two $x$ -intercepts and add them.

Step 1: Determine $a$ , $b$ and $c$
- $a = 3$ , $b = -12$ and $c = 5$ .
Step 2: Determine the sum of the roots
- The sum of the roots is $-\dfrac{b}{a} = -\dfrac{-12}{3}$ .
- $-(-12) = 12$ , so the sum is $\dfrac{12}{3} = 4$ .

The correct answer is

4

The same example in Desmos

new graph
Keys: y=3x^2 → -12x+5
Desmos shows: A parabola.
click the points where the curve meets the x-axis
Desmos shows: About $(0.472, 0)$ and $(3.528, 0)$ .
Desmos shows: $4$ : the sum of the two solutions.

Desmos expression list: y=3x^2-12x+5; 0.472+3.528 — Type

Desmos graph of y=3x^2-12x+5; 0.472+3.528, with (0.4724745, 0) and (3.5275255, 0) labelled — Look

Made with Desmos

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Your turn

$x^2 + bx - 24$ $= 0$

In the given equation, $b$ is a constant. One solution to the equation is $3$ . What is the other solution?

Show the answerHide the answer

Answer

The correct answer is $-8$ .

Step 1: Determine the product of the roots
- $a = 1$ and $c = -24$ .
- The product of the roots is $\dfrac{c}{a} = \dfrac{-24}{1} = -24$ .
Step 2: Determine the other root
- One root is $3$ . Call the other root $r$ : $3 \cdot r = -24$ .
- $r = \dfrac{-24}{3} = -8$ .

Traps

Dropping the minus sign in the sum. The sum is $-\dfrac{b}{a}$ . $\dfrac{b}{a} = \dfrac{-12}{3} = -4$ is a wrong answer.
Using $c$ for the sum. $\dfrac{c}{a} = \dfrac{5}{3}$ is the product of the roots, not their sum.

Practice this lesson36 questions

Unit 5: Parabolas in standard form

Key points

The graph of $y$ $= ax^2 + bx + c$ is a parabola.
If $a > 0$ , it opens upward and has a minimum. If $a < 0$ , it opens downward and has a maximum.
The $y$ -intercept is $(0, c)$ .
The vertex is at $x = -\dfrac{b}{2a}$ . Plug that $x$ back in to get its $y$ -coordinate.

See it solved

Example

The function $f$ is defined by $f(x)$ $= 2x^2 - 8x + 3$ . What is the minimum value of $f(x)$ ?

Worked by hand

The plan

Determine if the parabola opens upward or downward
Determine the $x$ -coordinate of the vertex
Determine the $y$ -coordinate of the vertex

Desmos: Desmos is quicker here: type the equation and click the vertex.

Step 1: Determine if the parabola opens upward or downward
- $a = 2$ , which is positive. The parabola opens upward.
- So the vertex is the lowest point, and its $y$ -coordinate is the minimum value.
Step 2: Determine the $x$ -coordinate of the vertex
- $a = 2$ and $b = -8$ .
- $x = -\dfrac{b}{2a} = -\dfrac{-8}{2 \cdot 2}$ .
- $2 \cdot 2 = 4$ , so $x = \dfrac{8}{4} = 2$ .
Step 3: Determine the $y$ -coordinate of the vertex
- $f(2)$ $= 2 \cdot 2^2 - 8 \cdot 2 + 3$ .
- $2^2 = 4$ , so $2 \cdot 4 = 8$ .
- $8 \cdot 2 = 16$ .
- $f(2)$ $= 8 - 16 + 3$ $= -5$ . The minimum value is $-5$ .

The correct answer is

-5

The same example in Desmos

new graph
Keys: y=2x^2 → -8x+3
Desmos shows: A parabola that opens upward.
click the lowest point
Desmos shows: $(2, -5)$ . The minimum value is $-5$ .

Desmos expression list: y=2x^2-8x+3 — Type

Desmos graph of y=2x^2-8x+3, with (2, −5) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $g$ is defined by $g(x)$ $= -x^2 + 6x + 4$ . What is the maximum value of $g(x)$ ?

Show the answerHide the answer

Answer

The correct answer is $13$ .

Step 1: Determine if the parabola opens upward or downward
- $a = -1$ , which is negative. The parabola opens downward, so the vertex is the highest point.
Step 2: Determine the $x$ -coordinate of the vertex
- $a = -1$ and $b = 6$ .
- $x$ $= -\dfrac{b}{2a}$ $= -\dfrac{6}{2 \cdot (-1)}$ .
- $2 \cdot (-1) = -2$ , so $x = -\dfrac{6}{-2} = 3$ .
Step 3: Determine the $y$ -coordinate of the vertex
- $g(3)$ $= -(3^2) + 6 \cdot 3 + 4$ .
- $3^2 = 9$ and $6 \cdot 3 = 18$ .
- $g(3)$ $= -9 + 18 + 4$ $= 13$ . The maximum value is $13$ .

Traps

Answering with the $x$ -coordinate. The vertex is at $x = 2$ , but the minimum value is the $y$ -coordinate, $-5$ . $2$ is a wrong answer.
Dropping a minus sign in $-\dfrac{b}{2a}$ . With $b = -8$ , $-b$ is $8$ . Using $-8$ gives $x = -2$ , and $f(-2)$ $= 8 + 16 + 3$ $= 27$ , a wrong answer.

Practice this lesson40 questions

Unit 6: Parabolas in context: height, profit and revenue

Key points

First decide which feature of the parabola the question asks for.
‘When is it highest?’ is the $x$ -coordinate of the vertex. ‘How high?’ is the $y$ -coordinate of the vertex.
The starting height is the $y$ -intercept: the value at time $0$ .
‘When does it land?’ is the positive $x$ -intercept, where the height is $0$ .

See it solved

Example

A ball is thrown upward. Its height, in feet, $t$ seconds after it is thrown is $h(t)$ $= -16t^2 + 64t + 5$ . What is the maximum height of the ball, in feet?

Worked by hand

The plan

Determine the feature asked for
Determine the $x$ -coordinate of the vertex
Determine the $y$ -coordinate of the vertex

Desmos: Desmos is quicker here: type the model, zoom out, and click the point the question asks for.

Step 1: Determine the feature asked for
- ‘Maximum height’ is the $y$ -coordinate of the vertex.
Step 2: Determine the $x$ -coordinate of the vertex
- $a = -16$ and $b = 64$ .
- $t$ $= -\dfrac{b}{2a}$ $= -\dfrac{64}{2 \cdot (-16)}$ .
- $2 \cdot (-16) = -32$ , so $t = -\dfrac{64}{-32} = 2$ seconds.
Step 3: Determine the $y$ -coordinate of the vertex
- Plug $t = 2$ into each term of $h(t)$ .
- First term: $2^2 = 4$ , so $-16 \cdot 4 = -64$ .
- Second term: $64 \cdot 2 = 128$ .
- The last term is $5$ .
- $-64 + 128 = 64$ , and $64 + 5 = 69$ .
- The maximum height is $69$ feet.

The correct answer is

69

The same example in Desmos

new graph
Keys: y=-16x^2 → +64x+5
Desmos shows: A parabola that opens downward, with $x$ for the seconds. Zoom out until the top is on screen.
click the highest point
Desmos shows: $(2, 69)$ : after $2$ seconds the ball is $69$ feet high.

Desmos expression list: y=-16x^2+64x+5 — Type

Desmos graph of y=-16x^2+64x+5, with (2, 69) labelled — Look

Made with Desmos

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Your turn

A model rocket is launched from the ground. Its height, in meters, $t$ seconds after launch is $h(t)$ $= -5t^2 + 30t$ . How many seconds after launch does the rocket land?

Show the answerHide the answer

Answer

The correct answer is $6$ .

Step 1: Determine the feature asked for
- The rocket lands when its height is $0$ . That is an $x$ -intercept.
Step 2: Set the height equal to $0$
- $-5t^2 + 30t = 0$ .
- Factor out $-5t$ : $-5t(t - 6) = 0$ .
- $-5t = 0$ gives $t = 0$ . $t - 6 = 0$ gives $t = 6$ .
- $t = 0$ is the launch. The rocket lands at $t = 6$ seconds.

Traps

Answering ‘how high’ with ‘when’. $t = 2$ is the time of the highest point. The height at that time is $69$ feet. $2$ is a wrong answer.
Reading the constant as the maximum. $5$ is the height at $t = 0$ , where the ball starts. It is not the greatest height.

Practice this lesson36 questions

Unit 7: Vertex form

Key points

Vertex form is $y$ $= a(x - h)^2 + k$ . The vertex is $(h, k)$ .
The sign inside the parentheses flips: $(x - 2)^2$ means $h = 2$ , and $(x + 2)^2$ means $h = -2$ .
$k$ is the minimum value if $a > 0$ , and the maximum value if $a < 0$ .
For the $y$ -intercept, plug in $x = 0$ .

See it solved

Example

The function $f$ is defined by $f(x)$ $= 3(x + 4)^2 - 7$ . In the $xy$ -plane, what is the vertex of the graph of $y = f(x)$ ?

A $(-4, -7)$
B $(-4, 7)$
C $(4, -7)$
D $(4, 7)$

Worked by hand

The plan

Determine the $x$ -coordinate of the vertex
Determine the $y$ -coordinate of the vertex

Desmos: Reading $h$ and $k$ takes one line. Desmos is a check: type the equation and click the vertex.

Step 1: Determine the $x$ -coordinate of the vertex
- Write $(x + 4)$ as $(x - (-4))$ . So $h = -4$ .
- That rules out choices C and D.
Step 2: Determine the $y$ -coordinate of the vertex
- $k$ is the number added at the end: $k = -7$ .
- The vertex is $(-4, -7)$ . That rules out choice B.

The correct answer choice is A:

(-4, -7)

The same example in Desmos

new graph
Keys: y=3(x+4)^2 → -7
Desmos shows: A parabola that opens upward.
click the lowest point
Desmos shows: $(-4, -7)$ : the vertex.

Desmos expression list: y=3(x+4)^2-7 — Type

Desmos graph of y=3(x+4)^2-7, with (−4, −7) labelled — Look

Made with Desmos

Open Desmos

Your turn

A parabola has its vertex at $(2, 3)$ and passes through the point $(4, 11)$ . Its equation is $y$ $= a(x - 2)^2 + 3$ , where $a$ is a constant. What is the value of $a$ ?

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Answer

The correct answer is $2$ .

Step 1: Plug the point into the equation
- Put $x = 4$ and $y = 11$ into $y$ $= a(x - 2)^2 + 3$ .
- $11$ $= a(4 - 2)^2 + 3$ .
- $4 - 2 = 2$ and $2^2 = 4$ . So $11 = 4a + 3$ .
Step 2: Solve for $a$
- Subtract $3$ from both sides: $4a = 11 - 3 = 8$ .
- Divide both sides by $4$ : $a = \dfrac{8}{4} = 2$ .

Traps

Keeping the sign inside the parentheses. $(x + 4)$ gives $h = -4$ , not $4$ . $(4, -7)$ is choice C, a wrong answer.
Flipping both signs. Only the number inside the parentheses flips. $k$ is read as written: $-7$ . $(-4, 7)$ is choice B, a wrong answer.

Practice this lesson12 questions

Unit 8: Factored form

Key points

Factored form is $y$ $= a(x - r)(x - s)$ . The $x$ -intercepts are $r$ and $s$ .
The vertex is halfway between them: its $x$ -coordinate is $\dfrac{r + s}{2}$ .
Plug that $x$ back in to get the vertex's $y$ -coordinate.
For the $y$ -intercept, plug in $x = 0$ .

See it solved

Example

The function $f$ is defined by $f(x)$ $= (x - 1)(x - 7)$ . What is the minimum value of $f(x)$ ?

Worked by hand

The plan

Determine the $x$ -intercepts
Determine the $x$ -coordinate of the vertex
Determine the $y$ -coordinate of the vertex

Desmos: Desmos is quicker here: type the equation and click the vertex or the intercepts.

Step 1: Determine the $x$ -intercepts
- $x - 1 = 0$ gives $x = 1$ .
- $x - 7 = 0$ gives $x = 7$ .
Step 2: Determine the $x$ -coordinate of the vertex
- The vertex is halfway between $1$ and $7$ .
- $\dfrac{1 + 7}{2} = \dfrac{8}{2} = 4$ .
Step 3: Determine the $y$ -coordinate of the vertex
- $f(4)$ $= (4 - 1)(4 - 7)$ .
- $4 - 1 = 3$ and $4 - 7 = -3$ .
- $f(4)$ $= 3 \cdot (-3)$ $= -9$ . The minimum value is $-9$ .

The correct answer is

-9

The same example in Desmos

new graph
Desmos shows: A parabola that opens upward.
click the points where the curve meets the x-axis
Desmos shows: $(1, 0)$ and $(7, 0)$ .
click the lowest point
Desmos shows: $(4, -9)$ . The minimum value is $-9$ .

Desmos expression list: y=(x-1)(x-7) — Type

Desmos graph of y=(x-1)(x-7), with (1, 0) and (7, 0) and (4, −9) labelled — Look

Made with Desmos

Open Desmos

Your turn

The function $g$ is defined by $g(x)$ $= -2(x + 3)(x - 5)$ . What is the maximum value of $g(x)$ ?

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Answer

The correct answer is $32$ .

Step 1: Determine the $x$ -intercepts
- $x + 3 = 0$ gives $x = -3$ .
- $x - 5 = 0$ gives $x = 5$ .
Step 2: Determine the $x$ -coordinate of the vertex
- $\dfrac{-3 + 5}{2} = \dfrac{2}{2} = 1$ .
Step 3: Determine the $y$ -coordinate of the vertex
- Plug $x = 1$ into each factor: $1 + 3 = 4$ and $1 - 5 = -4$ .
- $g(1)$ $= -2 \cdot 4 \cdot (-4)$ .
- $4 \cdot (-4) = -16$ , and $-2 \cdot (-16) = 32$ .
- The maximum value is $32$ .

Traps

Reading the intercepts with the wrong sign. $(x - 1)(x - 7)$ has intercepts $1$ and $7$ , not $-1$ and $-7$ . The midpoint of $-1$ and $-7$ is $-4$ , which gives the wrong vertex.
Stopping at the midpoint. $x = 4$ is where the minimum happens. The minimum value is $f(4) = -9$ . $4$ is a wrong answer.

Practice this lesson36 questions

Unit 9: Switching between forms

Key points

The same parabola can be written in three forms: standard, vertex and factored.
$a$ is the same number in all three forms.
Vertex form shows the vertex. Factored form shows the $x$ -intercepts. Standard form shows the $y$ -intercept.
To write vertex form, find the vertex with $x = -\dfrac{b}{2a}$ , then write $a(x - h)^2 + k$ .

See it solved

Example

Which equation is equivalent to $y = x^2 - 6x + 5$ and shows the coordinates of the vertex of its graph as constants?

A $y$ $= (x - 3)^2 - 4$
B $y$ $= (x - 3)^2 + 5$
C $y$ $= (x + 3)^2 - 4$
D $y$ $= (x - 1)(x - 5)$

Worked by hand

The plan

Determine the $x$ -coordinate of the vertex
Determine the $y$ -coordinate of the vertex

Desmos: With answer choices, Desmos is quicker: type the given equation, then each choice, and keep the one that lies exactly on top.

Step 1: Determine the $x$ -coordinate of the vertex
- $a = 1$ and $b = -6$ .
- $x$ $= -\dfrac{b}{2a}$ $= -\dfrac{-6}{2 \cdot 1}$ $= \dfrac{6}{2}$ $= 3$ . So $h = 3$ .
- Vertex form has $(x - 3)^2$ . That rules out choice C.
- Choice D is not in vertex form, so it does not show the vertex.
Step 2: Determine the $y$ -coordinate of the vertex
- Plug $x = 3$ into $x^2 - 6x + 5$ .
- $3^2 = 9$ and $6 \cdot 3 = 18$ .
- $9 - 18 + 5 = -4$ . So $k = -4$ .
- $y$ $= (x - 3)^2 - 4$ . That rules out choice B.

The correct answer choice is A:

y

= (x - 3)^2 - 4

The same example in Desmos

new graph
Keys: y=x^2 → -6x+5
Desmos shows: The parabola.
Keys: y=(x-3)^2 → -4
Desmos shows: Choice A lies exactly on top of it (dashed in the picture).
click the lowest point
Desmos shows: $(3, -4)$ : the vertex, which choice A shows as $3$ and $-4$ .

Desmos expression list: y=x^2-6x+5; y=(x-3)^2-4 — Type

Desmos graph of y=x^2-6x+5; y=(x-3)^2-4, with (3, −4) labelled — Look

Made with Desmos

Open Desmos

Your turn

The equation $y$ $= 2(x - 1)^2 - 8$ is rewritten in the form $y$ $= ax^2 + bx + c$ , where $a$ , $b$ and $c$ are constants. What is the value of $c$ ?

Show the answerHide the answer

Answer

The correct answer is $-6$ .

Step 1: Simplify the parentheses
- $(x - 1)^2$ $= (x - 1)(x - 1)$ $= x^2 - 2x + 1$ .
- Multiply by $2$ : $2x^2 - 4x + 2$ .
Step 2: Combine the constants
- After multiplying out, the constants are $2$ and $-8$ .
- $2 - 8 = -6$ . So $c = -6$ .
- The standard form is $2x^2 - 4x - 6$ .
- Another way: $c$ is the value of $y$ at $x = 0$ .
- $(0 - 1)^2 = 1$ , so $y$ $= 2 \cdot 1 - 8$ $= -6$ .

Traps

Using $c$ as $k$ . In $y = x^2 - 6x + 5$ , the $5$ is the $y$ -coordinate of the $y$ -intercept, not of the vertex. $(x - 3)^2 + 5$ is choice B, a wrong answer.
Picking an equivalent form that does not show what is asked. Choice D, $(x - 1)(x - 5)$ , is the same parabola, but it shows the $x$ -intercepts, not the vertex.

Practice this lesson12 questions

Unit 10: Where a line meets a parabola

Key points

At a point where two graphs meet, both equations are true.
Set the two expressions for $y$ equal, then move everything to one side.
Solve the quadratic for $x$ . Then use the line to get $y$ .
The discriminant of that quadratic counts the meeting points: positive means two, zero means one, negative means none.

See it solved

Example

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

The graphs of the given equations meet at the point $(x, y)$ , where $x > 0$ . What is the value of $y$ ?

Worked by hand

The plan

Equate the two expressions
Determine the roots
Determine the value of $y$

Desmos: Desmos is quicker here: type both equations as given and click the points where the graphs meet.

Step 1: Equate the two expressions
- Both equal $y$ : $x^2 - 2x - 3$ $= 2x + 2$ .
- Subtract $2x$ from both sides: $x^2 - 4x - 3 = 2$ .
- Subtract $2$ from both sides: $x^2 - 4x - 5 = 0$ .
Step 2: Determine the roots
- Two numbers that multiply to $-5$ and add to $-4$ : they are $-5$ and $1$ .
- $(x - 5)(x + 1)$ $= 0$ , so $x = 5$ or $x = -1$ .
- The question wants $x > 0$ , so $x = 5$ .
Step 3: Determine the value of $y$
- Use the line: $y$ $= 2 \cdot 5 + 2$ $= 10 + 2$ $= 12$ .
- Check in the parabola: $5^2 - 2 \cdot 5 - 3$ $= 25 - 10 - 3$ $= 12$ . It matches.

The correct answer is

12

The same example in Desmos

new graph
Keys: y=x^2 → -2x-3
Desmos shows: The parabola.
Desmos shows: The line. It meets the parabola twice.
click the two points where they meet
Desmos shows: $(-1, 0)$ and $(5, 12)$ . The one with $x > 0$ is $(5, 12)$ , so $y = 12$ .

Desmos expression list: y=x^2-2x-3; y=2x+2 — Type

Desmos graph of y=x^2-2x-3; y=2x+2, with (−1, 0) and (5, 12) labelled — Look

Made with Desmos

Open Desmos

Your turn

$y$ $= x^2 - 6x + 11$
$y = k$

In the given system, $k$ is a constant. The graphs of the two equations meet at exactly one point. What is the value of $k$ ?

Show the answerHide the answer

Answer

The correct answer is $2$ .

Step 1: Equate the two expressions
- $x^2 - 6x + 11$ $= k$ .
- Subtract $k$ from both sides: $x^2 - 6x + (11 - k)$ $= 0$ .
Step 2: Set the discriminant equal to $0$
- $a = 1$ , $b = -6$ and $c = 11 - k$ .
- One meeting point means $b^2 - 4ac = 0$ : $(-6)^2 - 4 \cdot 1 \cdot (11 - k)$ $= 0$ .
- $(-6)^2 = 36$ , and $4 \cdot (11 - k)$ $= 44 - 4k$ . So $36 - 44 + 4k = 0$ .
- $36 - 44 = -8$ , so $-8 + 4k = 0$ .
- $4k = 8$ , so $k = 2$ .

Traps

Stopping at $x$ . $x = 5$ is where the graphs meet, but the question asks for $y$ . Plug it in: $y = 12$ . $5$ is a wrong answer.
Using the wrong root. $x = -1$ is a meeting point too, but it is not positive. It gives $y$ $= 2 \cdot (-1) + 2$ $= 0$ , a wrong answer.

Practice this lesson96 questions

Unit 11: Matching coefficients

Key points

If two expressions are equal for every value of $x$ , their matching coefficients are equal.
Expand both sides into the same form first.
Then match the $x^2$ terms, the $x$ terms and the constants, one at a time.
A missing term has a coefficient of $0$ .

See it solved

Example

$(ax + 3)(2x - 5)$ $= 8x^2 + bx - 15$

The given equation is true for all values of $x$ , where $a$ and $b$ are constants. What is the value of $b$ ?

Worked by hand

The plan

Expand both sides into the same form
Equate the coefficients

Desmos: Match the coefficients by hand. Desmos is a check: graph both sides with your values. They should lie exactly on top of each other.

Step 1: Expand both sides into the same form
- FOIL the left side. First $ax \cdot 2x = 2ax^2$ and $ax \cdot (-5) = -5ax$ .
- Then $3 \cdot 2x = 6x$ and $3 \cdot (-5) = -15$ .
- The $x^2$ term is $2ax^2$ .
- The $x$ terms are $-5ax$ and $6x$ . Together they are $(-5a + 6)x$ .
- The constant is $-15$ .
Step 2: Equate the coefficients
- The $x^2$ terms: $2a = 8$ , so $a = 4$ .
- The $x$ terms: $-5a + 6 = b$ .
- $-5 \cdot 4 + 6$ $= -20 + 6$ $= -14$ . So $b = -14$ .
- The constants are $-15$ on both sides. They already match.

The correct answer is

-14

The same example in Desmos

new graph
Desmos shows: The left side with $a = 4$ .
Keys: y=8x^2 → -14x-15
Desmos shows: The right side with $b = -14$ . It lies exactly on top (dashed in the picture).

Desmos expression list: y=(4x+3)(2x-5); y=8x^2-14x-15 — Type

Desmos graph of y=(4x+3)(2x-5); y=8x^2-14x-15 — Look

Made with Desmos

Open Desmos

Your turn

$x^2 + 6x + c$ $= (x + d)^2$

The given equation is true for all values of $x$ , where $c$ and $d$ are constants. What is the value of $c$ ?

Show the answerHide the answer

Answer

The correct answer is $9$ .

Step 1: Expand both sides into the same form
- $(x + d)^2$ $= x^2 + 2dx + d^2$ .
Step 2: Equate the coefficients
- The $x$ terms: $2d = 6$ , so $d = 3$ .
- The constants: $c$ $= d^2$ $= 3^2$ $= 9$ .

Traps

Forgetting one of the two $x$ products. The $x$ term comes from $-5ax$ and from $6x$ . Using only $-5a$ gives $b = -20$ , a wrong answer.
Matching the wrong pair. $2a$ is the coefficient of $x^2$ , so it matches $8$ , not $b$ . Setting $2a = b$ mixes up two different terms.

Practice this lesson60 questions

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
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.
Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
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.
Start a new graph for each example: a letter or function defined twice gives the error 'You've defined … in more than one place'.

Start practicing

Practice by lesson

Lesson 1, Solving a quadratic, 56 questions
Lesson 2, Hidden quadratics, 32 questions
Lesson 3, How many solutions? The discriminant, 56 questions
Lesson 4, Sum and product of the roots, 36 questions
Lesson 5, Parabolas in standard form, 40 questions
Lesson 6, Parabolas in context: height, profit and revenue, 36 questions
Lesson 7, Vertex form, 12 questions
Lesson 8, Factored form, 36 questions
Lesson 9, Switching between forms, 12 questions
Lesson 10, Where a line meets a parabola, 96 questions
Lesson 11, Matching coefficients, 60 questions

Mixed practice: every lesson

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Next chapter · Chapter 7Absolute value

Desmos

Next set

1Unit 1: Solving a quadratic

Key points

See it solved

Worked by hand

Step 1: Determine two factors of acacac that add to bbb

Step 2: Split the value of bbb

Step 3: Factor by grouping

Step 4: Determine the roots

The same example in Desmos

Your turn

Step 1: Set the equation equal to 000

Step 2: Determine two factors of ccc that add to bbb

Step 3: Determine the roots

Traps

2Unit 2: Hidden quadratics

Key points

See it solved

Worked by hand

Step 1: Rename the repeated piece

Step 2: Determine the roots

Step 3: Substitute back

The same example in Desmos

Your turn

Step 1: Rename the repeated piece

Step 2: Determine the roots

Step 3: Substitute back

Traps

3Unit 3: How many solutions? The discriminant

Key points

See it solved

Worked by hand

Step 1: Set the discriminant equal to 000

Step 2: Solve for ccc

The same example in Desmos

Your turn

Step 1: Determine the number of solutions using the discriminant

Traps

4Unit 4: Sum and product of the roots

Key points

See it solved

Worked by hand

Step 1: Determine aaa, bbb and ccc

Step 2: Determine the sum of the roots

The same example in Desmos

Your turn

Step 1: Determine the product of the roots

Step 2: Determine the other root

Traps

5Unit 5: Parabolas in standard form

Key points

See it solved

Worked by hand

Step 1: Determine if the parabola opens upward or downward

Step 2: Determine the xxx-coordinate of the vertex

Step 3: Determine the yyy-coordinate of the vertex

The same example in Desmos

Your turn

Step 1: Determine if the parabola opens upward or downward

Step 2: Determine the xxx-coordinate of the vertex

Step 3: Determine the yyy-coordinate of the vertex

Traps

6Unit 6: Parabolas in context: height, profit and revenue

Key points

See it solved

Worked by hand

Step 1: Determine the feature asked for

Step 2: Determine the xxx-coordinate of the vertex

Step 3: Determine the yyy-coordinate of the vertex

The same example in Desmos

Your turn

Step 1: Determine the feature asked for

Step 2: Set the height equal to 000

Traps

7Unit 7: Vertex form

Key points

See it solved

Worked by hand

Step 1: Determine the xxx-coordinate of the vertex

Step 2: Determine the yyy-coordinate of the vertex

The same example in Desmos

Unit 1: Solving a quadratic

Step 1: Determine two factors of $ac$ that add to $b$

Step 2: Split the value of $b$

Step 1: Set the equation equal to $0$

Step 2: Determine two factors of $c$ that add to $b$

Unit 2: Hidden quadratics

Unit 3: How many solutions? The discriminant

Step 1: Set the discriminant equal to $0$

Step 2: Solve for $c$

Unit 4: Sum and product of the roots

Step 1: Determine $a$ , $b$ and $c$

Unit 5: Parabolas in standard form

Step 2: Determine the $x$ -coordinate of the vertex

Step 3: Determine the $y$ -coordinate of the vertex

Step 2: Determine the $x$ -coordinate of the vertex

Step 3: Determine the $y$ -coordinate of the vertex

Unit 6: Parabolas in context: height, profit and revenue

Step 2: Determine the $x$ -coordinate of the vertex

Step 3: Determine the $y$ -coordinate of the vertex

Step 2: Set the height equal to $0$

Unit 7: Vertex form

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

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

Step 2: Solve for $a$

Unit 8: Factored form

Step 1: Determine the $x$ -intercepts

Step 2: Determine the $x$ -coordinate of the vertex

Step 3: Determine the $y$ -coordinate of the vertex

Step 1: Determine the $x$ -intercepts

Step 2: Determine the $x$ -coordinate of the vertex

Step 3: Determine the $y$ -coordinate of the vertex

Unit 9: Switching between forms

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

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

Unit 10: Where a line meets a parabola

Step 3: Determine the value of $y$

Step 2: Set the discriminant equal to $0$

Unit 11: Matching coefficients