Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
(4 points)
To complete the square for the equation y = -x^2 + 8x + 7, we need to rewrite the equation in the form y = a(x - h)^2 + k.
First, we group the x terms:
y = -(x^2 - 8x) + 7
Next, we need to add and subtract the square of half the coefficient of x in the expression x^2 - 8x. Since the coefficient of x is -8, half of it is -4, and the square of -4 is 16. We add and subtract 16 inside the parentheses:
y = -(x^2 - 8x + 16 - 16) + 7
Now, we can rewrite the expression inside the parentheses as a perfect square:
y = -(x^2 - 8x + 16) + 16 + 7
Next, we simplify the expression -x^2 + 8x + 16 by rewriting it as -(x - 4)^2:
y = -(x - 4)^2 + 23
Now the equation is in the desired form y = a(x - h)^2 + k, where a = -1, h = 4, and k = 23.
The vertex of the parabola is given by the values (h, k), so the vertex is (4, 23).
The axis of symmetry is the vertical line that passes through the vertex, so the axis of symmetry is x = 4.
To complete the square and determine the vertex and axis of symmetry of the quadratic equation y = -x^2 + 8x + 7, follow these steps:
Step 1: Group the x terms and isolate the constant term:
y = -(x^2 - 8x) + 7
Step 2: To complete the square, take half of the coefficient of x (which is 8) and square it, then add and subtract the result within the parentheses:
y = -(x^2 - 8x + 16 - 16) + 7
Step 3: Rearrange the parentheses to create a perfect square trinomial:
y = -(x^2 - 8x + 16) -(-16) + 7
Step 4: Simplify the equation inside the parentheses and combine like terms:
y = -(x-4)^2 + 9
Step 5: Now, the quadratic equation is in vertex form: y = a(x-h)^2 + k, where (h, k) is the vertex. From this form, we can determine that the vertex is (h, k) = (4, 9).
Step 6: The axis of symmetry is a vertical line that passes through the vertex. In this case, the axis of symmetry is x = 4.
Therefore, the vertex is (4, 9) and the axis of symmetry is x = 4.
To complete the square and determine the vertex and axis of symmetry of the quadratic equation y = -x^2 + 8x + 7, follow these steps:
Step 1: Identify the coefficients
The given quadratic equation is in the standard form of a quadratic equation, which is y = ax^2 + bx + c. In this equation, a = -1, b = 8, and c = 7.
Step 2: Divide the coefficient of x by 2 and square the result
Take the coefficient of x (which is 8 in this case), divide it by 2, and then square the result. (8/2)^2 = 16.
Step 3: Add and subtract the value obtained in Step 2 within the equation
Add and subtract the value obtained in Step 2 within the quadratic equation:
y = -x^2 + 8x + 7 + 16 - 16
Step 4: Rearrange the terms
Rearrange the terms in the equation by grouping the common terms:
y = (-x^2 + 8x + 16) - 16 + 7
Step 5: Factorize the perfect square trinomial
Factorize the perfect square trinomial obtained in Step 4:
y = (-(x - 4)^2) - 9
Step 6: Rewrite in vertex form
Rewrite the equation in vertex form by completing the square:
y = -(x - h)^2 + k
From Step 5, we have y = -(x - 4)^2 - 9. Comparing this to the vertex form, we can see that the vertex is (h, k) = (4, -9).
Step 7: Determine the axis of symmetry
The axis of symmetry is given by the equation x = h. Therefore, in this case, the axis of symmetry is x = 4.
So, the vertex of the parabola is (4, -9) and the axis of symmetry is x = 4.