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: Recall slope-intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope (which determines the steepness of the line) and \(b\) is the \(y\) - intercept (the value of \(y\) when \(x = 0\)).

For the function \(f(x)=\frac{1}{4}x - 1\), the slope \(m_f=\frac{1}{4}\) and the \(y\) - intercept \(b_f=- 1\).

For the function \(g(x)=\frac{1}{2}x - 2\), the slope \(m_g=\frac{1}{2}\) and the \(y\) - intercept \(b_g=-2\).

Step2: Compare the slopes

To compare the steepness, we compare the magnitudes of the slopes. Since \(\frac{1}{2}>\frac{1}{4}\) (because \(\frac{1}{2}=\frac{2}{4}\) and \(\frac{2}{4}>\frac{1}{4}\)), a larger slope means a steeper line. So the line of \(g(x)\) is steeper than the line of \(f(x)\) because \(m_g>m_f\).

Step3: Compare the \(y\) - intercepts

To compare the \(y\) - intercepts, we look at the values of \(b\). We have \(b_f=-1\) and \(b_g = - 2\). Since \(-2<-1\) (on the number line, - 2 is to the left of - 1), the \(y\) - intercept of \(g(x)\) is lower than the \(y\) - intercept of \(f(x)\).

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.)