Question

timmy writes the equation $f(x) = \frac{1}{4}x - 1$. he then doubles both of the terms on the right side to create the equation $g(x) = \frac{1}{2}x - 2$. how does the graph of $g(x)$ compare to the graph of $f(x)$?
\bigcirc the line of $g(x)$ is steeper and has a higher $y$-intercept.
\bigcirc the line of $g(x)$ is less steep and has a lower $y$-intercept.
\bigcirc the line of $g(x)$ is steeper and has a lower $y$-intercept.
\bigcirc the line of $g(x)$ is less steep and has a higher $y$-intercept.

Explanation:

Step1: Analyze slopes

For a linear function \( y = mx + b \), \( m \) is the slope. For \( f(x)=\frac{1}{4}x - 1 \), slope \( m_f=\frac{1}{4} \). For \( g(x)=\frac{1}{2}x - 2 \), slope \( m_g=\frac{1}{2} \). Since \( \frac{1}{2}>\frac{1}{4} \), \( g(x) \) has a steeper slope.

Step2: Analyze y - intercepts

The y - intercept is \( b \) in \( y = mx + b \). For \( f(x) \), \( b_f=- 1 \). For \( g(x) \), \( b_g=-2 \). Since \( -2 < - 1 \), \( g(x) \) has a lower y - intercept.

Answer:

The line of \( g(x) \) is steeper and has a lower y - intercept. (Corresponding to the option: The line of \( g(x) \) is steeper and has a lower y - intercept.)