Question

What are the vertex and x-intercepts of the graph of the function y = (x - 4)(x + 2)?

a) Vertex: (1, -9); Intercepts: x = 4, -2
b) Vertex: (4, -2); Intercepts: x = -4, 2
c) Vertex: (-4, 2); Intercepts: x = 3, -3
d) Vertex: (1, -5); Intercepts: x = -4, 2

Answer 1

The vertex of the quadratic function y = (x - 4)(x + 2) is (1, -9), and the x-intercepts are x = 4 and x = -2, which corresponds to option (a).

Step-by-step explanation:

To find the vertex and x-intercepts of the quadratic function y = (x - 4)(x + 2), we first identify its x-intercepts by setting y to zero and solving for x:

• y = (x - 4)(x + 2) = 0
• x - 4 = 0 or x + 2 = 0
• x = 4 or x = -2

These are the x-intercepts of the graph. Next, the vertex of a parabola given by y = ax2 + bx + c is at the point (h, k), where h = -b/(2a). In the given equation, if we expand it we have:

• y = x2 - 2x - 8

The coefficients are a = 1 and b = -2. Therefore, h = -(-2)/(2*1) = 1. To find k, we substitute h into the original equation:

• y = (1 - 4)(1 + 2)
• y = (-3)(3)
• y = -9

So the vertex is (1, -9). Considering our findings, the correct options for the vertex and x-intercepts are:

• Vertex: (1, -9)
• Intercepts: x = 4, -2

Option (a) is the correct one: Vertex: (1, -9); Intercepts: x = 4, -2.