Question

the function f(x) is shown on the graph.
what is the equation of f(x)?

Explanation:

Step1: Identify the vertex and intercepts

Assume each grid square has side length 1. The graph is a piecewise linear function (a V - shaped graph, absolute - value - like). The vertex (the peak) seems to be at (-2, 4) (we can infer from the symmetry and the intercepts), and it passes through the origin (0,0) and another x - intercept, say (-4, 0)? Wait, no, looking at the graph, let's re - evaluate. Wait, the left part: let's find two points on the left line. Let's say when x=-4, y = 0? Wait, no, the left line crosses the x - axis at, let's assume the vertex is at (-2, 4). Wait, maybe it's better to use the two segments. The function is a piecewise function with two linear parts: one increasing and one decreasing.

Let's find the slope of the left segment. Let's take two points: suppose the left segment goes from (-4, 0) to (-2, 4). The slope $m_1=\frac{4 - 0}{-2-(-4)}=\frac{4}{2}=2$. The equation of the left line using point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(-4,0)$: $y-0 = 2(x + 4)\Rightarrow y = 2x+8$.

For the right segment, it goes from (-2, 4) to (0,0). The slope $m_2=\frac{0 - 4}{0-(-2)}=\frac{-4}{2}=-2$. The equation of the right line using point - slope form with $(x_1,y_1)=(0,0)$: $y-0=-2(x - 0)\Rightarrow y=-2x$.

But we can also express this as an absolute - value function. The general form of an absolute - value function is $y=a|x - h|+k$, where $(h,k)$ is the vertex. If the vertex is at $(-2,4)$, then $y=a|x + 2|+4$. Since it passes through (0,0), substitute $x = 0,y = 0$:

$0=a|0 + 2|+4\Rightarrow0 = 2a+4\Rightarrow2a=-4\Rightarrow a=-2$.

So the equation is $y=-2|x + 2|+4$. Let's simplify this:

$y=-2|x + 2|+4=-2(x + 2)+4$ when $x\geq - 2$ (but wait, when $x\geq - 2$, the expression inside the absolute value is non - negative, so $|x + 2|=x + 2$; when $x\lt - 2$, $|x + 2|=-(x + 2)$).

Simplify for $x\geq - 2$: $y=-2x-4 + 4=-2x$.

For $x\lt - 2$: $y=-2(-(x + 2))+4=2x + 4+4=2x + 8$.

Let's check the origin: when $x = 0$, $y=-2(0)=0$, which matches. When $x=-4$, $y=2(-4)+8=0$, which matches the x - intercept. When $x=-2$, $y=-2(-2)=4$ (from the right segment) or $y=2(-2)+8 = 4$ (from the left segment), so the vertex is at (-2,4).

Alternatively, maybe the graph is $y=-2|x + 2|+4$, which can be rewritten as $y=-2|x+2| + 4$. But let's check the slope again. Wait, maybe the vertex is at (-2, 4), and the function is $f(x)=-2|x + 2|+4$. Let's expand it:

$f(x)=-2|x + 2|+4$. When $x\geq - 2$, $f(x)=-2(x + 2)+4=-2x-4 + 4=-2x$. When $x\lt - 2$, $f(x)=-2(-(x + 2))+4=2x + 4+4=2x + 8$.

Another way: Let's use the two - point form for each line.

Left line: passes through (-4,0) and (-2,4). Slope $m=\frac{4 - 0}{-2-(-4)} = 2$. Equation: $y-0=2(x + 4)\Rightarrow y = 2x+8$.

Right line: passes through (-2,4) and (0,0). Slope $m=\frac{0 - 4}{0-(-2)}=-2$. Equation: $y-0=-2(x - 0)\Rightarrow y=-2x$.

So the piecewise function is:

$f(x)=\begin{cases}2x + 8, & x\lt - 2\\-2x, & x\geq - 2\end{cases}$

We can also write this as an absolute - value function. The general form of an absolute - value function is $y=a|x - h|+k$. The vertex is at $(h,k)=(-2,4)$. We know that when $x = 0$, $y = 0$. Substitute into $y=a|x+2|+4$:

$0=a|0 + 2|+4\Rightarrow2a=-4\Rightarrow a=-2$. So $y=-2|x + 2|+4$, which is equivalent to the piecewise function above.

Step2: Verify the equations

For the left part ($x\lt - 2$), take $x=-3$: $y=2(-3)+8=2$, and $y=-2|-3 + 2|+4=-2| - 1|+4=-2 + 4=2$. Correct.

For the right part ($x\geq - 2$), take $x = 1$: $y=-2(1)=-2$, and $y=-2|1 + 2|+4=-2(3)+4=-6 + 4=-2$. Correct.

And at $x=-2$, $y=-2|-2 + 2|+4=4$, which is the vertex. At $x = 0$, $y = 0$, which matches the graph. At $x=-4$, $y=-2|-4 + 2|+4=-2(2)+4=0$, which matches the x - intercept.

Answer:

The equation of the function $f(x)$ is $f(x)=

$$\begin{cases}2x + 8, & x\lt - 2\\-2x, & x\geq - 2\end{cases}$$

$ or in absolute - value form $f(x)=-2|x + 2|+4$. If we simplify the absolute - value form:

$f(x)=-2|x + 2|+4=-2(x + 2)+4=-2x-4 + 4=-2x$ (for $x\geq - 2$) and $f(x)=2x + 8$ (for $x\lt - 2$). A more compact form (if we consider the graph's symmetry and the fact that it can be represented as an absolute - value function) is $f(x)=-2|x + 2|+4$ or the piecewise linear function as above. If we assume the vertex is at (-2,4) and use the two - segment approach, the equation is $f(x)=

$$\begin{cases}2x + 8, & x\lt - 2\\-2x, & x\geq - 2\end{cases}$$

$.