Question

complete the description of the piecewise function graphed below.
$f(x)=\

$$\begin{cases}\\quad\\text{if } -6\\leq x\\leq -1\\\\\\quad\\text{if } -1< x\\leq 2\\\\\\quad\\text{if } 2< x\\leq 6\\end{cases}$$

$
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Explanation:

Step1: Find first segment's equation

We use the two points $(-6, 2)$ and $(-1, -1)$ for the line where $-6 \leq x \leq -1$. First calculate slope:
$m = \frac{-1 - 2}{-1 - (-6)} = \frac{-3}{5} = -\frac{3}{5}$
Use point-slope form $y - y_1 = m(x - x_1)$ with $(-6,2)$:
$y - 2 = -\frac{3}{5}(x + 6)$
Simplify:
$y = -\frac{3}{5}x - \frac{18}{5} + 2 = -\frac{3}{5}x - \frac{8}{5}$

Step2: Find second segment's equation

For $-1 < x \leq 2$, the graph is a horizontal line at $y = -2$, so the function is $y = -2$.

Step3: Find third segment's equation

Use the two points $(2, -3)$ and $(6, 2)$ for the line where $2 < x \leq 6$. Calculate slope:
$m = \frac{2 - (-3)}{6 - 2} = \frac{5}{4}$
Use point-slope form $y - y_1 = m(x - x_1)$ with $(2, -3)$:
$y + 3 = \frac{5}{4}(x - 2)$
Simplify:
$y = \frac{5}{4}x - \frac{10}{4} - 3 = \frac{5}{4}x - \frac{11}{2}$

Answer:

$$ f(x) =
LATEXBLOCK0
$$