Unit 7

Units 1 to 6 Review

textbook

Part 7.2B corresponds to section 2.1, page 62, section 5.2, page 238, section 7.2, page 346, and section 9.1, page 430 of your Pre-Calculus 12 textbook.

Radical Functions

Radical functions can be compared to

y = \sqrt{x}

for graphing purposes when considering transformations.

The function

y = a \sqrt{b (x - h)} + k

shows the following transformations.

a vertical stretch about the $x$ -axis by a factor of $|a|$
a vertical reflection across the $x$ -axis if $a$ is negative
a horizontal stretch about the $y$ -axis by a factor of $\frac{1}{|b|}$
a horizontal reflection across the $y$ -axis if $b$ is negative
a horizontal translation of $h$ units
a vertical translation of $k$ units

Sinusoidal Functions

The sine and cosine functions

y = \sin θ

and

y = \cos θ

can also be transformed.

Consider

y = a \sin b (θ - c) + d

and

y = a \cos b (θ - c) + d

, where

$|a|$ is the amplitude, or vertical stretch factor,

$|a| = \frac{\max - \min}{2}$

$\frac{1}{|b|}$ is the horizontal stretch factor. The $period = \frac{2 π}{|b|}$ for both sine and cosine functions,
$c$ is the phase shift, or horizontal translation, and
$d$ is the vertical displacement, or the vertical translation.

$d = \frac{\max + \min}{2}$

Exponential Functions

The exponential function

y = c^{x}

can be transformed to

y = a {(c)}^{b (x - h)} + k

, where

$|a|$ is the vertical stretch factor, and a negative $a$ corresponds to a vertical reflection across the $x$ -axis,
$\frac{1}{|b|}$ is the horizontal stretch factor, and a negative $b$ corresponds to a horizontal reflection across the $y$ -axis,
$h$ is the horizontal translation, and
$k$ is the vertical translation.

Rational Functions

The rational function

y = \frac{1}{x}

can also be transformed.

Consider the function

y = \frac{a}{(x - h)} + k

, where

$|a|$ is the vertical stretch factor, and a negative $a$ corresponds to a vertical reflection across the $x$ -axis,
$h$ is the horizontal translation, and
$k$ is the vertical translation.