Algebra Examples

g (x) = - \frac{1}{2} f (x)

$g (x) = - \frac{1}{2} f (x)$

Step 1

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

f (x) = - \frac{1}{2} \cdot f (x)

$f (x) = - \frac{1}{2} \cdot f (x)$

Step 2

Find the y-intercepts.

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Step 2.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.2

Using the slope-intercept form, find the y-intercept for

f (x) = - \frac{1}{2} \cdot f (x)

$f (x) = - \frac{1}{2} \cdot f (x)$ .

b_{1} = 0

$b_{1} = 0$

Step 2.3

Using the slope-intercept form, find the y-intercept for

g (x) = - \frac{1}{2} \cdot f (x)

$g (x) = - \frac{1}{2} \cdot f (x)$ .

b_{2} = 0

$b_{2} = 0$

Step 2.4

List the y-intercepts.

b_{1} = 0

$b_{1} = 0$

b_{2} = 0

$b_{2} = 0$

b_{1} = 0

$b_{1} = 0$

b_{2} = 0

$b_{2} = 0$

Step 3

The vertical shift depends on the y-intercept value

b

$b$ , where

b = b_{2} - b_{1}

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

b_{2} - b_{1} = 0

$b_{2} - b_{1} = 0$

Step 4

Since

b = 0

$b = 0$ , the graph is not shifted.

Not Shifted

Step 5

Find the slopes.

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Step 5.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 5.2

Using the slope-intercept form, find the slope for

f (x) = - \frac{1}{2} \cdot f (x)

$f (x) = - \frac{1}{2} \cdot f (x)$ .

m_{1} = - \frac{1}{2}

$m_{1} = - \frac{1}{2}$

Step 5.3

Using the slope-intercept form, find the slope for

g (x) = - \frac{1}{2} \cdot f (x)

$g (x) = - \frac{1}{2} \cdot f (x)$ .

m_{2} = - \frac{1}{2}

$m_{2} = - \frac{1}{2}$

Step 5.4

List the slopes.

m_{1} = - \frac{1}{2}

$m_{1} = - \frac{1}{2}$

m_{2} = - \frac{1}{2}

$m_{2} = - \frac{1}{2}$

m_{1} = - \frac{1}{2}

$m_{1} = - \frac{1}{2}$

m_{2} = - \frac{1}{2}

$m_{2} = - \frac{1}{2}$

Step 6

The vertical stretch depends on the slope.

| m_{2} | < | m_{1} |

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

| m_{2} | > | m_{1} |

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

| m_{2} | = | m_{1} |

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

Step 7

Since

| m_{2} | = | m_{1} |

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

No vertical stretch or compression

Step 8

Since

m_{1}

$m_{1}$ and

m_{2}

$m_{2}$ 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

f (x) = - \frac{1}{2} \cdot f (x)

$f (x) = - \frac{1}{2} \cdot f (x)$ .

No transformation

Step 10