1. **Problem Statement:** We are given three modular functions in the form $y = |ax + b|$ with vertices at $(1,0)$ and different slopes for each function: $f(x)$ with slopes $\pm 2$, $g(x)$ with slopes $\pm 1$, and $h(x)$ with slopes $\pm 0.5$. 2. **General form and vertex:** The vertex form of a modular function is $y = |a(x - h)|$ where $(h,0)$ is the vertex. Here, the vertex is at $(1,0)$, so the functions can be written as: $$y = |a(x - 1)|$$ 3. **Relating slope to $a$:** The slope of the arms of the V-shaped graph corresponds to the absolute value of $a$. The left arm has slope $-a$ and the right arm has slope $+a$. 4. **Finding $a$ and $b$ for each function:** Since the vertex form is $y = |a(x - 1)| = |ax - a|$, we can rewrite it as: $$y = |ax + b|$$ where $b = -a$. - For $f(x)$ with slopes $\pm 2$: - $a = 2$ - $b = -2$ - So, $f(x) = |2x - 2|$ - For $g(x)$ with slopes $\pm 1$: - $a = 1$ - $b = -1$ - So, $g(x) = |x - 1|$ - For $h(x)$ with slopes $\pm 0.5$: - $a = 0.5$ - $b = -0.5$ - So, $h(x) = |0.5x - 0.5|$ 5. **Summary:** - $f(x) = |2x - 2|$ with $a=2$, $b=-2$ - $g(x) = |x - 1|$ with $a=1$, $b=-1$ - $h(x) = |0.5x - 0.5|$ with $a=0.5$, $b=-0.5$