Question

consider the function $f(x)$ graphed below.
write a formula for $f(x)$.
$f(x) = \square$

Explanation:

Step1: Identify vertex of the piecewise function

The vertex (corner point) of the graph is at $(-4, -2)$. This is the point where the slope of the line changes.

Step2: Find equation for $x \geq -4$

First, identify two points on this segment: $(-4, -2)$ and $(0, 2)$.
Calculate slope: $m = \frac{2 - (-2)}{0 - (-4)} = \frac{4}{4} = 1$.
Use slope-intercept form $y = mx + b$, substitute $m=1$ and $(0,2)$: $b=2$.
Equation: $y = x + 2$ for $x \geq -4$.

Step3: Find equation for $x < -4$

Identify two points on this segment: $(-4, -2)$ and $(-10, 4)$.
Calculate slope: $m = \frac{4 - (-2)}{-10 - (-4)} = \frac{6}{-6} = -1$.
Use point-slope form $y - y_1 = m(x - x_1)$ with $(-4, -2)$:
$y - (-2) = -1(x - (-4))$
Simplify: $y + 2 = -x - 4 \implies y = -x - 6$ for $x < -4$.

Step4: Combine as absolute value function

Since this is a V-shaped graph, we can rewrite the piecewise function as an absolute value function.
The general form is $f(x) = a|x - h| + k$, where $(h,k)$ is the vertex $(-4, -2)$.
The slope of the right segment is $a=1$, so substitute:
$f(x) = |x - (-4)| + (-2) = |x + 4| - 2$

Answer:

$f(x) = |x + 4| - 2$
or in piecewise form:

$$ f(x)=
LATEXBLOCK0
$$