Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7.
To complete the square and determine the vertex and axis of symmetry, follow these steps:
Step 1: Write the equation in the form y = ax^2 + bx + c. In this case, we have y = -x^2 + 8x + 7.
Step 2: Take out the common factor of -1 from the quadratic term: y = -(x^2 - 8x) + 7.
Step 3: To complete the square, take half of the coefficient b (in this case, 8) and square it. Add that value inside the parentheses, and subtract it outside the parentheses to keep the equation balanced. In this case, the value is (8/2)^2 = 16.
So the equation becomes: y = -(x^2 - 8x + 16 - 16) + 7.
Step 4: Simplify the equation inside the parentheses: y = -(x^2 - 8x + 16) + 16 + 7.
Step 5: Factor the trinomial inside the parentheses by completing the square: y = -((x - 4)^2 - 16) + 23.
Step 6: Simplify further: y = -(x - 4)^2 + 23.
The equation is now in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Comparing the equation to the vertex form, we can see that the vertex is at (h, k) = (4, 23).
Step 7: The axis of symmetry is given by x = h, so in this case, the axis of symmetry is x = 4.
Therefore, the vertex of the parabola is (4, 23), and the axis of symmetry is x = 4.
To complete the square for the given quadratic function y = -x^2 + 8x + 7, we need to manipulate the quadratic equation and rewrite it in the form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
First, let's rewrite the equation:
y = -x^2 + 8x + 7
To complete the square, we need to add and subtract (-8/2)^2 = 16 to the right side of the equation, since (-8/2)^2 = 16. However, since we added 16 to the right side, we also need to subtract 16 from the right side to ensure the equation remains balanced.
y + 16 - 16 = -x^2 + 8x + 7
Rearranging the quadratic terms:
y = -(x^2 - 8x) + 7 + 16 - 16
Now, we can rewrite the quadratic term as a perfect square:
y = -(x^2 - 8x + 16) + 7 + 16 - 16
Simplifying the equation inside the parentheses:
y = -(x - 4)^2 + 7
The equation is now in the desired form y = a(x - h)^2 + k.
Comparing this to the general equation form, we can identify the values of a, h, and k. We find:
a = -1, h = 4, k = 7.
Therefore, the vertex of the parabola is at the point (4, 7), and the axis of symmetry is the vertical line x = 4.
To complete the square for the quadratic equation y = -x^2 + 8x + 7, follow these steps:
Step 1: Rearrange the equation
Start by moving the constant term (7) to the other side of the equation:
y - 7 = -x^2 + 8x
Step 2: Group the x-terms
Rearrange the right side of the equation to group the x-terms together:
y - 7 = -(x^2 - 8x)
Step 3: Perform the square completion
To complete the square for the expression inside the parentheses, focus on the coefficient of the x-term, which is 8. Divide this coefficient by 2 and square it:
(8/2)^2 = 16
Step 4: Add and subtract the square completion value
Add and subtract the square completion value (16) inside the parentheses:
y - 7 = -(x^2 - 8x + 16 - 16)
Step 5: Simplify the equation
Simplify the expression inside the parentheses by factoring it as a perfect square:
y - 7 = -(x - 4)^2 + 16
Step 6: Write the equation in vertex form
Distribute the negative sign to the terms inside the parentheses:
y - 7 = -x^2 + 8x - 16 + 16
Simplify the equation:
y - 7 = -x^2 + 8x
Move the constant term (7) to the other side of the equation:
y = -x^2 + 8x + 7
The equation is now in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, the vertex is (h, k) = (4, 7).
Therefore, the vertex of the parabola is (4, 7).
Step 7: Determine the axis of symmetry
The axis of symmetry of a parabola is the vertical line that passes through the vertex. The equation of the axis of symmetry is given by x = h.
In this case, the axis of symmetry is x = 4.
Therefore, the vertex is (4, 7) and the axis of symmetry is x = 4.