Stretches and compressions; even and odd functions

Warm-up

Three quick questions from earlier lessons.

Discover

Move the slider and watch the graph of $y = (bx)^2$ .

The graph of y = (bx)², a parabola that gets narrower as b grows.

When $b$ goes from 1 to 2, the graph becomes…

narrower
wider

Key idea

Multiplying stretches or squeezes a graph. Symmetry has a name.

For

a > 1

:

y = a f(x)

stretches the graph vertically by

a

. For

b > 1

:

y = f(bx)

compresses it horizontally by

b

.

f

is even if

f(-x) = f(x)

(symmetric about the

y

-axis), odd if

f(-x) = -f(x)

(symmetric about the origin).

Worked example

Is $f(x) = x^3 - 2x$ even, odd, or neither?

f(-x) = (-x)^3 - 2(-x)

Replace x by (−x).

= -x^3 + 2x

(−x)³ = −x³ and −2(−x) = 2x.

= -(x^3 - 2x) = -f(x)

: odd

Factor out −1: this is −f(x).

Your turn, guided

Your turn: $f(x) = x^4 + 3$ . Check one value.

f(-1) = (-1)^4 + 3

Replace x by −1.

f(-1) = 4

(−1)⁴ = 1. It equals f(1): a sign that f is even.

Practice ★

How does the graph of $y = f(4x)$ compare with the graph of $y = f(x)$ ?
1. It is squeezed toward the y-axis: horizontal compression by a factor of 4
2. It is pulled away from the y-axis: horizontal stretch by a factor of 4
3. It is stretched vertically by a factor of 4
How does the graph of $y = f(5x)$ compare with the graph of $y = f(x)$ ?
1. It is squeezed toward the y-axis: horizontal compression by a factor of 5
2. It is pulled away from the y-axis: horizontal stretch by a factor of 5
3. It is stretched vertically by a factor of 5
How does the graph of $y = f(3x)$ compare with the graph of $y = f(x)$ ?
1. It is squeezed toward the y-axis: horizontal compression by a factor of 3
2. It is pulled away from the y-axis: horizontal stretch by a factor of 3
3. It is stretched vertically by a factor of 3

Watch out!

Practice ★★

$f(x) = 8x^3 - 9x$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither
$f(x) = 9x^3 - 2x$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither

$f(x) = 3x^4 - 8x^2 + 7$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither
$f(x) = 5x^4 - 1x^2 + 9$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither

Exit ticket

Three questions to check you have got it.

$f(x) = 8x^3 - 9x$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither
$f(x) = 9x^3 - 2x$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither
$f(x) = 7x^3 - 1x$ . Is $f$ even, odd, or neither?
1. Even
2. Odd
3. Neither

Summary card

Stretches, even and odd

$a f(x)$ : vertical stretch by $a$ . $f(bx)$ : horizontal compression by $b$ .
Even: $f(-x) = f(x)$ , mirror image in the $y$ -axis.
Odd: $f(-x) = -f(x)$ , symmetric about the origin.

Watch out: compute $f(-x)$ ; the signs of the coefficients do not decide.

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