Final answer:
The x-coordinate of the vertex is -1 and the y-coordinate is -9. The x-intercepts are (1, 0) and (-4, 0).
Step-by-step explanation:
To find the x-coordinate of the vertex, we can use the formula {x = -b/2a}. In the quadratic equation y = x² + 2x - 8, the coefficient of x² is 1, the coefficient of x is 2, and the constant term is -8. Plugging these values into the formula gives us x = -2/2(1) = -1. To find the y-coordinate of the vertex, we substitute the value of x (-1) into the quadratic equation and solve for y. Plugging x = -1 into the equation gives us y = (-1)² + 2(-1) - 8 = 1 - 2 - 8 = -9. Therefore, the coordinates of the vertex are (-1, -9).
To find the x-intercepts, or the values of x where y = 0, we set the quadratic equation equal to 0 and solve for x. Setting y = 0 in the equation gives us x² + 2x - 8 = 0. We can solve this equation using factoring, completing the square, or the quadratic formula. Using the quadratic formula, we have x = (-2 ± √(2² - 4(1)(-8))) / (2(1)) = (-2 ± √(4 + 32)) / 2 = (-2 ± √36) / 2 = (-2 ± 6) / 2. This gives us the solutions x = (-2 + 6) / 2 = 2/2 = 1 and x = (-2 - 6) / 2 = -8/2 = -4. Therefore, the x-intercepts are (1, 0) and (-4, 0).