William W. answered • 10/28/21
Top Pre-Calc Tutor
y = -2x2 - 2x + 7
Step 1: Move the constant term off to the side:
y = -2x2 - 2x + 7
Step 2: Factor out the coefficient in front of the "x" (which is -2):
y = -2(x2 + x) + 7
Step 3: Using the coefficient in front of the "x" term, which is "1", divide it in half (which is 1/2) and then square it (which is 1/4) and then add the result to the expression in the parenthesis HOWEVER, to balance things, subtract the same thing outside the parenthesis (making sure to include the factor out front):
y = -2(x2 + x + 1/4) - (-2(1/4)) + 7
Step 4: Write the expression inside the parenthesis as a "square" and combine the terms on the outside of the parenthesis:
y = -2(x + 1/2)2 + 15/2
This is the vertex form (y = a(x - h)2 + k) although your question infers it is "standard form". Standard form is usually y = ax2 + bx + c. You'll also need to tweak this a bit depending on how your book or teacher define what they want as "standard form".
To find the y-intercept, make the "x" equal to zero:
It's easier to do this with the original form y = -2x2 - 2x + 7:
y = -2(0)2 - 2(0) + 7 = 7 so the y-intercept is (0, 7)
To find the x-intercept, make the "y" equal to zero:
0 = -2(x + 1/2)2 + 15/2
-15/2 = -2(x + 1/2)2
15/4 = (x + 1/2)2
(x + 1/2) = ±√(15/4) = ±√15/2
x = -1/2 ±√15/2
x = (-1 ± √15)/2
The vertex is the (h, k) from the vertex form y = a(x - h)2 + k
So, since the vertex form is y = -2(x + 1/2)2 + 15/2 then the vertex is (-1/2, 15/2)
The line of symmetry is x = -1/2
Since the coefficient outside the parenthesis ("a") is negative (it's "-2"), then the vertex is a maximum.
The graph looks like this:
