Final answer:
The x-intercepts of the graph of the equation y=(x-4)(x+2) are at (4,0) and (-2,0). The vertex of this parabola is at (1,-9), which is found by averaging the x-intercepts and substituting this value back into the equation.
Step-by-step explanation:
To find the vertex and x-intercepts of the graph of the quadratic equation y=(x-4)(x+2), we need to first determine the x-intercepts by setting y to zero and solving for x:
- 0 = (x-4)(x+2)
- x - 4 = 0 or x + 2 = 0
- x = 4 or x = -2
Therefore, the x-intercepts are at (4,0) and (-2,0). To find the vertex, we use the fact that the vertex lies exactly halfway between the x-intercepts for a parabola in standard form. The average of 4 and -2 is 1, so the x-coordinate of the vertex is 1. To find the y-coordinate of the vertex, we substitute x = 1 into the equation:
- y = (1-4)(1+2)
- y = (-3)(3)
- y = -9
The vertex is at (1,-9). Therefore, the correct answers are: X-intercepts: (4,0), (-2,0) and Vertex: (1,-9).