Question

which of the following is the axis of symmetry for the function shown below? (graph of a parabola) options: a) x=3, b) x=7, c) x=6, d) x=2, e) x=-2

Explanation:

Step1: Identify x-intercepts

The graph intersects the x-axis at \( x = -2 \) and \( x = 6 \).

Step2: Calculate midpoint

The axis of symmetry for a parabola is the midpoint of its x-intercepts. The midpoint formula is \( \frac{x_1 + x_2}{2} \). Substituting \( x_1 = -2 \) and \( x_2 = 6 \), we get \( \frac{-2 + 6}{2} = \frac{4}{2} = 2 \)? Wait, no, wait. Wait, looking at the graph again, maybe I misread the x-intercepts. Wait, the left x-intercept is at \( x = -2 \)? Wait, no, the grid: the left intersection is at \( x = -2 \)? Wait, no, the x-axis: the left root is at \( x = -2 \)? Wait, no, the graph: let's check the x-intercepts. The left one is at \( x = -2 \)? Wait, no, the right one is at \( x = 6 \)? Wait, no, the options are x=3,7,6,2,-2. Wait, maybe the x-intercepts are at \( x = -2 \) and \( x = 6 \)? Wait, no, let's recalculate. Wait, midpoint of -2 and 6 is \( (-2 + 6)/2 = 4/2 = 2 \)? But option D is x=2, but the first option A is x=3. Wait, maybe I misread the x-intercepts. Wait, the graph: the left x-intercept is at \( x = -2 \)? No, wait, the grid lines: the x-axis has marks at -4, -2, 0, 2, 4, 6. So the left intersection is at \( x = -2 \), right at \( x = 6 \). Then midpoint is \( (-2 + 6)/2 = 2 \). But option D is x=2. But wait, maybe the x-intercepts are at \( x = -2 \) and \( x = 6 \)? Wait, no, maybe I made a mistake. Wait, another way: the vertex is at x=2? Wait, the graph's peak is at x=2? Wait, the y-axis is 0, then x=2 is the middle. Wait, the options: A is x=3, D is x=2. Wait, let's check again. Wait, maybe the x-intercepts are at \( x = -2 \) and \( x = 6 \), midpoint is 2, so axis is x=2. So the answer is D? Wait, but the first option A is x=3. Wait, maybe I misread the x-intercepts. Wait, maybe the left x-intercept is at \( x = -2 \) and the right at \( x = 6 \), midpoint is 2. So the axis of symmetry is x=2, which is option D. Wait, but let's check the graph again. The peak is at x=2? The graph goes up to x=2? Wait, the y-axis is 0, then x=2 is the vertical line through the vertex. So the axis of symmetry is x=2, so option D.

Answer:

D. \( x = 2 \)