Question

what function does this graph represent?
a. $f(x) = 0.5(x + 4)^2 - 5$
b. $f(x) = -0.5(x - 4)^2 - 5$
c. $f(x) = 0.5(x - 4)^2 - 5$
d. $f(x) = -0.5(x + 4)^2 - 5$

Explanation:

Step1: Analyze the parabola's direction and vertex

The parabola opens upwards (since the coefficient of the squared term should be positive), so we can eliminate options B and D (their coefficients are -0.5, which would open downwards). Now we have options A and C. The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the graph, the vertex seems to be at \((4, -5)\) (since the minimum point is around x = 4, y = -5). For option A, the vertex would be \((-4, -5)\) (because \( h = -4 \) from \( (x + 4) = (x - (-4)) \)), which doesn't match. For option C, the vertex is \((4, -5)\) (since \( h = 4 \) from \( (x - 4) \)), which matches the graph's vertex. Also, the coefficient \( a = 0.5 \) is positive, so the parabola opens upwards, consistent with the graph.

Step2: Verify with the y-intercept (optional but confirmatory)

Let's check the y-intercept (x = 0) for option C: \( f(0) = 0.5(0 - 4)^2 - 5 = 0.5(16) - 5 = 8 - 5 = 3 \). Looking at the graph, the y-intercept is around 3 (between 2 and 4), which matches. For option A: \( f(0) = 0.5(0 + 4)^2 - 5 = 0.5(16) - 5 = 8 - 5 = 3 \)? Wait, no, wait: Wait, option A is \( 0.5(x + 4)^2 - 5 \), so at x=0, it's \( 0.5(16) -5 = 8 -5 = 3 \), same as C? Wait, maybe my initial vertex analysis was wrong. Wait, no, looking at the graph, the vertex is at (4, -5), so the h should be 4, so the equation should be \( a(x - 4)^2 -5 \). So option C is \( 0.5(x - 4)^2 -5 \), which has vertex (4, -5). The graph's vertex is at (4, -5) (the minimum point is at x=4, y=-5), so that's correct. Option A has vertex (-4, -5), which is on the left side, but the graph's vertex is on the right side (x=4), so option C is correct.

Answer:

C. \( f(x) = 0.5(x - 4)^2 - 5 \)