Question

determine the effects on the graph of the parent function f(x) = x for the g(x) = 7f(x) function. graph both functions on the same coordinate grid.

determine the effects on the graph of the parent function f(x) = x for the g(x) = 7f(x) function.

a. the graph becomes steeper
b. the graph becomes less steep

select the correct graph of both the functions.

a.
graph with two lines, one steeper
b.
graph with two lines, less steep difference
c.
graph with negative slope lines
d.
graph with negative slope lines

the slope of the graph of g(x) is 3 times the slope of the graph of f(x).

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x) = x \), which has a slope of \( 1 \) (since in the form \( y = mx + b \), \( m = 1 \) and \( b = 0 \)).

Step2: Analyze the transformed function

The transformed function is \( g(x) = 7f(x) \). Substituting \( f(x) = x \), we get \( g(x) = 7x \). The slope of \( g(x) \) is \( 7 \).

Step3: Compare the slopes

A larger slope (in absolute value) means the line is steeper. Since \( 7>1 \), the slope of \( g(x) \) is greater than the slope of \( f(x) \). So the graph of \( g(x) \) becomes steeper than the graph of \( f(x) \).

Step4: Analyze the graph options

• For the graph of \( f(x)=x \) (blue) and \( g(x)=7x \) (red):
• The line \( y = x \) has a slope of \( 1 \), and \( y = 7x \) has a slope of \( 7 \). The line with a larger slope is steeper. Looking at the options, option A shows the red line ( \( g(x) \)) steeper than the blue line ( \( f(x) \)), which matches our analysis.

Step5: Slope multiple

The slope of \( f(x) \) is \( 1 \) and the slope of \( g(x) \) is \( 7 \). So the slope of \( g(x) \) is \( 7 \) times the slope of \( f(x) \) (the original text had a typo, it should be \( 7 \) instead of \( 3 \)).

Answer:

For the effect on the graph: A. The graph becomes steeper.

For the correct graph: A (the graph with the red line steeper than the blue line, passing through points consistent with \( y = 7x \) and \( y = x \)).

For the slope multiple: The slope of the graph of \( g(x) \) is \( 7 \) times the slope of the graph of \( f(x) \).