Final Answer:
The quadratic function in standard form is f(x) = x² + 2x - 8. The vertex of the function is (-1, -9), and the x- and y-intercepts are (-4, 0) and (0, -8) respectively. The graph of the function opens upwards.
Step-by-step explanation:
To express the given quadratic function, (f(x) = x² + 2x - 8), in standard form, we complete the square. Starting with the given function, we add and subtract ((2/2)² = 1) inside the parentheses:
f(x) = x² + 2x - 8 + 1 - 1
This allows us to rewrite the expression as a perfect square trinomial:
f(x) = (x + 1)² - 9
Now, the function is in standard form, (f(x) = a(x - h)² + k), where (h, k) is the vertex. In this case, the vertex is (-1, -9).
To find the x-intercepts, we set(f(x) to zero and solve for x:
(x + 1)² - 9 = 0
Solving this equation gives x = -4 and x = 2. Therefore, the x-intercepts are (-4, 0) and (2, 0).
For the y-intercept, we set (x) to 0:
f(0) = (0 + 1)^2 - 9 = -8
So, the y-intercept is (0, -8).
The graph of the function is a parabola that opens upwards, with the vertex at (-1, -9), x-intercepts at (-4, 0) and (2, 0), and a y-intercept at (0, -8).