Precalculus Examples

$f (x) = 4 x - 5$

Step 1

The parent function is the simplest form of the type of function given.

$g (x) = x$

Step 2

Find the y-intercepts.

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Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Using the slope-intercept form, find the y-intercept for $g (x) = x$ .

$b_{1} = 0$

Step 2.3

Using the slope-intercept form, find the y-intercept for $f (x) = 4 x - 5$ .

$b_{2} = - 5$

Step 2.4

List the y-intercepts.

$b_{1} = 0$

$b_{2} = - 5$

$b_{1} = 0$

$b_{2} = - 5$

Step 3

The vertical shift depends on the y-intercept value $b$ , where $b = b_{2} - b_{1}$

$b_{2} - b_{1} = - 5$

Step 4

Since $b < 0$ , the graph is shifted down $5$ units.

Shifted down $5$ units

Step 5

Find the slopes.

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Step 5.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 5.2

Using the slope-intercept form, find the slope for $g (x) = x$ .

$m_{1} = 1$

Step 5.3

Using the slope-intercept form, find the slope for $f (x) = 4 x - 5$ .

$m_{2} = 4$

Step 5.4

List the slopes.

$m_{1} = 1$

$m_{2} = 4$

$m_{1} = 1$

$m_{2} = 4$

Step 6

The vertical stretch depends on the slope.

If $| m_{2} | < | m_{1} |$ , vertical compression

If $| m_{2} | > | m_{1} |$ , vertical stretch

If $| m_{2} | = | m_{1} |$ , no vertical stretch or compression.

Step 7

Since $| m_{2} | > | m_{1} |$ , the graph is vertically stretched.

Vertical stretch

Step 8

Since $m_{1}$ and $m_{2}$ do not have opposite signs, the graph is not reflected about the y-axis.

Not reflected about the y-axis

Step 9

Describe the transformation from the function $g (x) = x$ .

Shifted down $5$ units

Vertical stretch

Step 10

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