Question

the graph of $y = f(x)$ is shown below. find the value of $f(-2).$

Explanation:

Step1: Understand the problem

We need to find the value of \( f(-2) \) from the graph of \( y = f(x) \). This means we look for the \( y \)-coordinate when \( x = -2 \) on the graph.

Step2: Locate \( x = -2 \) on the x - axis

Find the point on the x - axis where \( x=-2 \). Then, move vertically until we intersect the graph of \( y = f(x) \).

Step3: Determine the y - coordinate

From the graph, when we move up (or down) from \( x = -2 \) to the graph, we see that the graph passes through the point \( (-2, 0) \)? Wait, no, wait. Wait, let's check again. Wait, the graph is a parabola. Wait, when \( x=-2 \), let's see the grid. Wait, the x - axis has marks at -2, -1, 0, 1, 2, etc. The graph: let's see, when \( x = -2 \), the graph is at \( y = 0 \)? Wait, no, wait the vertex is at (0, -2). Wait, let's re - examine. The x - axis is horizontal, y - axis vertical. For \( x=-2 \), we find the vertical line \( x = -2 \) and see where it meets the graph. Looking at the graph, when \( x=-2 \), the graph intersects the x - axis? Wait, no, wait the graph crosses the x - axis at \( x=-1 \)? No, wait the graph: let's see the points. Wait, the vertex is at (0, -2). Then, when \( x = -2 \), let's calculate? Wait, maybe the equation of the parabola is \( y=\frac{1}{2}x^{2}-2 \)? Wait, no, let's check the graph. Wait, when \( x = -2 \), let's look at the grid. Each square is 1 unit. So, at \( x=-2 \), moving up from \( x=-2 \) on the x - axis, the graph is at \( y = 0 \)? Wait, no, maybe I made a mistake. Wait, let's look again. The graph: when \( x=-2 \), the vertical line \( x = -2 \) intersects the graph at \( y = 0 \)? Wait, no, wait the graph is a parabola opening upwards with vertex at (0, -2). Let's find the equation. The general form of a parabola with vertex at (h, k) is \( y=a(x - h)^{2}+k \). Here, h = 0, k=-2, so \( y = ax^{2}-2 \). When \( x = 2 \), \( y = 0 \) (since the graph crosses the x - axis at x = 2? Wait, no, the graph crosses the x - axis at x = 1? Wait, no, the graph: looking at the x - axis, the graph crosses the x - axis at x = -1? No, wait the graph in the picture: the x - axis has marks from -10 to 10. The graph touches the x - axis? Wait, no, the graph crosses the x - axis at x = -√4? No, wait, let's do it visually. At \( x=-2 \), the vertical line \( x=-2 \) intersects the graph at \( y = 0 \). Wait, maybe I was wrong earlier. Let's see: the graph is a parabola. When \( x=-2 \), the point on the graph is ( - 2, 0). Wait, but let's check again. Wait, the vertex is at (0, -2). So the equation is \( y=\frac{1}{2}x^{2}-2 \). Let's plug \( x=-2 \): \( y=\frac{1}{2}(-2)^{2}-2=\frac{1}{2}(4)-2 = 2 - 2=0 \). Yes, so \( f(-2)=0 \).

Answer:

\( 0 \)