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1. Quadratic patterns: constant second differences

Warm-up

Three quick questions from earlier lessons.

Discover

Squares of tiles: 1, 4, 9, 16 tiles. The differences are 3, 5, 7. What comes next in the pattern?

Key idea

Quadratic functions have constant second differences.

A quadratic function has the form

f(x) = ax^2 + bx + c

with

a \neq 0

. Over equal steps in

x

, its first differences change, but its second differences are constant (equal to

2a

for steps of 1). Linear: constant first differences; exponential: constant ratios.

Worked example

Values 2, 5, 10, 17, 26.

3, 5, 7, 9

First differences.

2, 2, 2: quadratic

Second differences.

a = 1

f(x) = x^2 + 1

for

x = 1, 2, …

2a = 2.

Your turn, guided

Your turn: values 0, 3, 12, 27, 48. The second difference.

6

Differences 3, 9, 15, 21.

Practice ★

A tile pattern has 6, 9, 14 and 21 tiles in steps 1 to 4 (a square of tiles plus 5 extra). How many tiles are in step 10?
A tile pattern has 10, 13, 18 and 25 tiles in steps 1 to 4 (a square of tiles plus 9 extra). How many tiles are in step 10?

For whole numbers $x$ from 1 to 14, which quantity is a quadratic function of $x$ ?
1. The area of a square with side $x$ cm
2. The perimeter of a square with side $x$ cm
3. The number of cells after $x$ doublings

Watch out!

Practice ★★

For $x = 0, 1, 2, 3, 4$ a function takes the values 5, 8, 13, 20, 29. What is the (constant) second difference?
For $x = 0, 1, 2, 3, 4$ a function takes the values 0, 9, 28, 57, 96. What is the (constant) second difference?

Exit ticket

Three questions to check you have got it.

A tile pattern has 6, 9, 14 and 21 tiles in steps 1 to 4 (a square of tiles plus 5 extra). How many tiles are in step 10?
A tile pattern has 10, 13, 18 and 25 tiles in steps 1 to 4 (a square of tiles plus 9 extra). How many tiles are in step 10?
A tile pattern has 5, 8, 13 and 20 tiles in steps 1 to 4 (a square of tiles plus 4 extra). How many tiles are in step 10?

Summary card

Quadratic patterns

Constant second differences mean quadratic.

Watch out: growing differences are not linear.

2. Standard and factored form of a quadratic

Warm-up

Three quick questions from earlier lessons.

Discover

$y = (x - 3)(x + 5)$ . Where does the graph cross the $x$ -axis?

At $x = 3$ and $x = -5$
At $x = -3$ and $x = 5$

Key idea

Two forms of the same quadratic.

Standard form

ax^2 + bx + c

shows the

y

-intercept

c

. Factored form

a(x - p)(x - q)

shows the

x

-intercepts

p

and

q

. Expand (multiply every term by every term) to go from factored to standard form.

Worked example

$(x + 4)(x - 2)$ .

x^2 - 2x + 4x - 8

x·x, x·(−2), 4·x, 4·(−2).

x^2 + 2x - 8

Collect.

x

-intercepts

-4

and

2

From the factors.

Your turn, guided

Your turn: expand $(x + 3)(x + 5)$ .

x^2 + 8x + 15

x² + 5x + 3x + 15.

Practice ★

Find the $x$ -intercepts of $y = (x - 6)(x + 2)$ .
Find the $x$ -intercepts of $y = (x - 7)(x + 1)$ .

Write $(x + 2)(x - 3)$ in standard form $ax^2 + bx + c$ .

Watch out!

Practice ★★

A quadratic with leading coefficient 1 crosses the $x$ -axis at $x = 9$ and $x = -4$ . Write it in factored form.
A quadratic with leading coefficient 1 crosses the $x$ -axis at $x = 7$ and $x = -6$ . Write it in factored form.

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercepts of $y = (x - 6)(x + 2)$ .
Find the $x$ -intercepts of $y = (x - 7)(x + 1)$ .
Find the $x$ -intercepts of $y = (x - 3)(x + 5)$ .

Summary card

Standard and factored form

Factored form gives the $x$ -intercepts; standard form gives the $y$ -intercept.

Watch out: the factor $x + 5$ gives the intercept $-5$ .

3. Graphs of quadratic functions: direction, intercepts, symmetry

Warm-up

Three quick questions from earlier lessons.

Discover

$y = -2x^2 + 8x + 3$ . What happens far to the left and right?

$y$ goes down on both sides
$y$ goes up on both sides

Key idea

Read a parabola from $y = ax^2 + bx + c$ .

a > 0

: opens up (minimum);

a < 0

: opens down (maximum); larger

|a|

, narrower.

c

is the $y$ -intercept. The vertex lies on the axis of symmetry, halfway between the

x

-intercepts.

Worked example

$y = x^2 - 6x + 5 = (x - 1)(x - 5)$ .

Opens up

a = 1 > 0.

y

-intercept 5

x = 0.

Vertex at

x = 3

y = -4

Halfway between 1 and 5.

Your turn, guided

Your turn: a parabola crosses the $x$ -axis at $-2$ and $6$ . The $x$ -coordinate of its vertex.

2

(−2 + 6)/2.

Practice ★

What is the $y$ -intercept of $y = 2x^2 + 1x + 11$ ?
What is the $y$ -intercept of $y = 4x^2 + 5x + 7$ ?

Does the graph of $y = -6x^2 + 5x + 2$ open up or down?
1. Down, so it has a maximum
2. Up, so it has a minimum
3. It depends on 2

Watch out!

Practice ★★

A parabola crosses the $x$ -axis at $x = -1$ and $x = 3$ . What is the $x$ -coordinate of its vertex?
A parabola crosses the $x$ -axis at $x = -9$ and $x = 5$ . What is the $x$ -coordinate of its vertex?

Boss problem. A ball is kicked from the ground. Its height in feet after $t$ seconds is $h(t) = -16t^2 + 32t = -16t(t - 2)$ .

Exit ticket

Three questions to check you have got it.

What is the $y$ -intercept of $y = 2x^2 + 1x + 11$ ?
What is the $y$ -intercept of $y = 4x^2 + 5x + 7$ ?
What is the $y$ -intercept of $y = 2x^2 + 6x + 15$ ?

Summary card

Graphs of quadratics

Sign of $a$ : up or down; $c$ : $y$ -intercept; vertex midway between the zeros.

Watch out: the $y$ -intercept is $c$ , not $a$ .

4. Vertex form of a quadratic

Warm-up

Three quick questions from earlier lessons.

Discover

$y = (x - 4)^2 + 1$ . A square is never negative. What is the smallest value of $y$ , and for which $x$ ?

$y = 1$ at $x = 4$
$y = 1$ at $x = -4$

Key idea

Vertex form shows the vertex.

In vertex form

y = a(x - h)^2 + k

the vertex is

(h, k)

. Compared with

y = ax^2

, the graph is shifted

h

right and

k

up. Watch the sign:

(x + 3)^2

means

h = -3

Worked example

$y = 2(x + 3)^2 - 5$ .

h = -3

x + 3 = x − (−3).

k = -5

Constant.

Vertex

(-3, -5)

, a minimum

a = 2 > 0.

Your turn, guided

Your turn: the $y$ -coordinate of the vertex of $y = -(x - 2)^2 + 9$ .

9

k = 9.

Practice ★

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?

What is the smallest value of $y = (x - 9)^2 - 3$ ?

Watch out!

Practice ★★

Write a quadratic with vertex $(8, 1)$ and leading coefficient 2, in vertex form.
Write a quadratic with vertex $(4, 1)$ and leading coefficient 4, in vertex form.

Exit ticket

Three questions to check you have got it.

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?
What is the $x$ -coordinate of the vertex of $y = 8(x + 5)^2 + 9$ ?

Summary card

Vertex form

$y = a(x - h)^2 + k$ : vertex $(h, k)$ .

Watch out: $(x + h)$ means the vertex is at $-h$ .

5. Quadratic functions: vertex and extreme value

Warm-up

Three quick questions from earlier lessons.

Discover

A ball's height is $h(t) = -5t^2 + 20t$ meters after $t$ seconds. Its graph is a parabola that opens downward. When does the ball reach its highest point?

After 2 seconds
After 4 seconds

Key idea

Every quadratic $f(x) = ax^2 + bx + c$ has a vertex, its highest or lowest point.

The vertex is at

x = -\dfrac{b}{2a}

. Its height

f\left(-\dfrac{b}{2a}\right)

is the smallest value of

f

a > 0

, the largest if

a < 0

Completing the square gives the vertex form $f(x) = a(x - h)^2 + k$ , with vertex $(h, k)$ .

Worked example

Find the vertex of $f(x) = 2x^2 - 12x + 5$ .

x = -\dfrac{-12}{2 \cdot 2} = 3

a = 2, b = −12.

f(3) = 18 - 36 + 5 = -13

Evaluate f at the vertex.

Vertex

(3, -13)

, minimum value

-13

a = 2 > 0, so this is the lowest point.

Your turn, guided

Your turn: $f(x) = x^2 + 6x + 11$ .

x = -\dfrac{6}{2} = -3

a = 1, b = 6.

Minimum value:

f(-3) = 2

f(−3) = 9 − 18 + 11.

Practice ★

$f(x) = 5x^2 - 10x + 20$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 40x + 10$ . What is the $x$ -coordinate of the vertex?
$f(x) = 2x^2 - 20x + 19$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 30x + 5$ . What is the $x$ -coordinate of the vertex?

Watch out!

Practice ★★

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . What is the smallest value of $f(x)$ ?

Draw the graph of $y = -(x - h)^2 + 6$ for $h = 0$ : place the vertex and one more point.
Draw the graph of $y = -(x - h)^2 + 2$ for $h = 2$ : place the vertex and one more point.

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . What is the smallest value of $f(x)$ ?

Summary card

Quadratic functions

Vertex at $x = -\dfrac{b}{2a}$ .
Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ( $a > 0$ ) or maximum ( $a < 0$ ).
Vertex form: $a(x - h)^2 + k$ , vertex $(h, k)$ .

Watch out: the constant term $c$ is $f(0)$ , not the extreme value.

6. Piecewise-defined functions and absolute value

Warm-up

Three quick questions from earlier lessons.

Discover

A parking lot charges 5 dollars for up to 2 hours, and 9 dollars for a longer stay. What does a stay of exactly 2 hours cost?

5 dollars
9 dollars

Key idea

A piecewise function uses a different formula on different parts of its domain.

To find

f(a)

: first find which condition

a

satisfies, then use that piece's formula. Check the boundary carefully (

<

\le

The absolute value is piecewise: $|x| = x$ if $x \ge 0$ , and $|x| = -x$ if $x < 0$ .

Worked example

$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x + 3 & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .

Use the second piece,

2x + 3

Check the conditions first: 1 is not less than 1, but 1 ≥ 1.

f(1) = 2 \cdot 1 + 3 = 5

Replace x by 1.

Your turn, guided

Your turn, with the same $f$ : find $f(-2)$ .

Use the first piece,

x^2

−2 < 1.

f(-2) = 4

(−2)² = 4.

Practice ★

$f(x) = \begin{cases} x + 4 & \text{if } x < 8 \\ 4x & \text{if } x \ge 8 \end{cases}$ . Find $f(0)$ .
$f(x) = \begin{cases} x + 8 & \text{if } x < 7 \\ 4x & \text{if } x \ge 7 \end{cases}$ . Find $f(2)$ .
$f(x) = \begin{cases} x + 7 & \text{if } x < 3 \\ 3x & \text{if } x \ge 3 \end{cases}$ . Find $f(1)$ .
$f(x) = \begin{cases} x + 6 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . Find $f(1)$ .

Watch out!

Practice ★★

$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . Find $f(3)$ .
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .
$f(x) = \begin{cases} x + 2 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .

For $x < 7$ , which expression is equal to $|x - 7|$ ?
1. $7 - x$
2. $x - 7$
3. $-x - 7$
For $x < 5$ , which expression is equal to $|x - 5|$ ?
1. $5 - x$
2. $x - 5$
3. $-x - 5$

Exit ticket

Three questions to check you have got it.

$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . Find $f(5)$ .
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . Find $f(3)$ .
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . Find $f(1)$ .

Summary card

Piecewise functions

Find the piece whose condition the input satisfies.
Use that piece's formula.
$|x| = x$ for $x \ge 0$ , and $-x$ for $x < 0$ .

Watch out: at a boundary, read $<$ and $\ge$ carefully.

7. Transformations: shifts and reflections

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = (x - h)^2 + k$ .

The graph of y = (x − h)² + k, a parabola whose vertex is at (h, k) and moves with the sliders.

When $h$ goes from 0 to 2, which way does the graph move?

To the right
To the left

Key idea

Changing the rule moves or flips the graph without changing its shape.

y = f(x - h) + k

moves the graph of

f

h

units to the right and

k

units up.

y = -f(x)

reflects it across the

x

-axis;

y = f(-x)

reflects it across the

y

-axis.

A change inside $f(\dots)$ acts on $x$ (horizontally); a change outside acts on $y$ (vertically).

Worked example

Describe $y = (x + 1)^2 - 3$ compared with $y = x^2$ .

1 unit to the left

x + 1 = x − (−1): h = −1.

3 units down

− 3 outside: k = −3.

Vertex:

(-1, -3)

The vertex (0, 0) moves with the graph.

Your turn, guided

Your turn: $y = (x - 4)^2 + 2$ .

4 units to the right

x − 4: h = 4.

2 units up

+ 2 outside.

The vertex is at

x = 4

The vertex moves from x = 0 to x = …

Practice ★

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 7 units to the right and 1 units down. Write the equation of the new graph.

Watch out!

Practice ★★

Draw the graph of $y = (x - 1)^2 - 3$ : place the vertex and one more point.
Draw the graph of $y = (x - 1)^2 - 1$ : place the vertex and one more point.

Match each equation with what it does to the graph of $y = f(x)$ .
1. $y = f(x) + 3$
2. $y = f(x - 3)$
3. $y = -f(x)$
4. $y = f(-x)$
1. moves it up 3 units
2. moves it 3 units to the right
3. reflects it across the x-axis
4. reflects it across the y-axis

Exit ticket

Three questions to check you have got it.

The graph of $y = x^2$ is moved 1 units to the right and 6 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 3 units to the right and 1 units down. Write the equation of the new graph.
The graph of $y = x^2$ is moved 4 units to the right and 6 units down. Write the equation of the new graph.

Summary card

Shifts and reflections

$f(x - h)$ : $h$ units right. $f(x) + k$ : $k$ units up.
$-f(x)$ : flip across the $x$ -axis. $f(-x)$ : flip across the $y$ -axis.
Inside $f(\dots)$ acts on $x$ ; outside acts on $y$ .

Watch out: $f(x + 3)$ moves the graph left.

8. Inverse functions

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = x + 7$ adds 7. Which function undoes it, and gives back the number you started with?

$x - 7$
$\dfrac{1}{x + 7}$
$x + 7$

Key idea

The inverse function $f^{-1}$ undoes $f$ : it swaps inputs and outputs.

f(a) = b

, then

f^{-1}(b) = a

. To find

f^{-1}(x)

: write

y = f(x)

, solve for

x

, then rename. The graph of

f^{-1}

is the reflection of the graph of

f

across the line

y = x

$f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ : the $-1$ is not an exponent here.

Worked example

$f(x) = 3x - 5$ . Find $f^{-1}(x)$ .

y = 3x - 5

Name the output y.

y + 5 = 3x

f subtracts 5 last, so undo that first: add 5.

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

Then undo × 3: divide by 3.

f^{-1}(x) = \dfrac{x + 5}{3}

Rename the input x.

Your turn, guided

Your turn: $f(x) = 2x + 6$ .

y = 2x + 6

Name the output y.

y - 6 = 2x

Undo + 6 first.

f^{-1}(x) = \dfrac{x - 6}{2}

Divide by 2, then rename.

Practice ★

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(x)$ .

Watch out!

Practice ★★

$f(x) = 2x + 6$ . Find $f^{-1}(10)$ .
$f(x) = 2x + 9$ . Find $f^{-1}(11)$ .
$f(x) = 3x + 1$ . Find $f^{-1}(10)$ .

The point $(2, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .
The point $(5, 7)$ is on the graph of $f$ . Tap the point that must be on the graph of $f^{-1}$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 7x + 9$ . Find $f^{-1}(x)$ .
$f(x) = 2x + 2$ . Find $f^{-1}(x)$ .
$f(x) = 7x + 5$ . Find $f^{-1}(x)$ .

Summary card

Inverse functions

Write $y = f(x)$ .
Solve for $x$ , undoing the last operation first.
Rename: $f^{-1}(x)$ . Check that $f(f^{-1}(x)) = x$ .

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

Answers: Quadratic functions and modeling

1. Quadratic patterns: constant second differences

$105$
$109$

The area of a square with side $x$ cm

$2$
$10$

$105$
$109$
$104$

2. Standard and factored form of a quadratic

$6,\ -2$
$7,\ -1$

$x^{2} - x - 6$

$\left(x - 9\right) \left(x + 4\right)$
$\left(x - 7\right) \left(x + 6\right)$

$6,\ -2$
$7,\ -1$
$3,\ -5$

3. Graphs of quadratic functions: direction, intercepts, symmetry

$11$
$7$

Down, so it has a maximum

$1$
$-2$

$16$

$11$
$7$
$15$

4. Vertex form of a quadratic

$-1$
$-9$

$-3$

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

$-1$
$-9$
$-5$

5. Quadratic functions: vertex and extreme value

$-4$
$18$
$-6$

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

$-4$
$18$
$-6$

6. Piecewise-defined functions and absolute value

$4$
$10$
$8$
$7$

$15$
$6$
$5$
$10$

$7 - x$
$5 - x$

$15$
$6$
$5$

7. Transformations: shifts and reflections

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$
$\left(x - 7\right)^{2} - 1$

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

$y = f(x) + 3$ → moves it up 3 units · $y = f(x - 3)$ → moves it 3 units to the right · $y = -f(x)$ → reflects it across the x-axis · $y = f(-x)$ → reflects it across the y-axis

$\left(x - 1\right)^{2} - 6$
$\left(x - 3\right)^{2} - 1$
$\left(x - 4\right)^{2} - 6$

8. Inverse functions

$\frac{x}{7} - \frac{9}{7}$
$\frac{x}{2} - 1$
$\frac{x}{7} - \frac{5}{7}$
$\frac{x}{2} - \frac{9}{2}$

$(7, 2)$
$(7, 5)$

$\frac{x}{7} - \frac{9}{7}$
$\frac{x}{2} - 1$
$\frac{x}{7} - \frac{5}{7}$