Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7 .(4 points)
To complete the square, we need to rewrite the given equation in the form y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.
First, let's group the terms with x:
y = -x^2 + 8x + 7
= -(x^2 - 8x) + 7
Now, we want to find a constant term to complete the square. To do this, we need to take half of the coefficient of x, square it, and add it to both sides of the equation:
y = -(x^2 - 8x + 16) + 7 + 16
= -(x^2 - 8x + 16) + 23
Note that we added +16 to the right side because we added 16 inside the parentheses, so we need to balance it out. We also added +16 inside the parentheses because (x - 4)^2 = x^2 - 8x + 16.
Now, we can rewrite the equation as:
y = -(x - 4)^2 + 23
Comparing this equation with y = a(x-h)^2 + k, we can see that the vertex is (h,k) = (4, 23).
The axis of symmetry is the vertical line passing through the vertex, so the axis of symmetry is x = 4.
To complete the square for the quadratic equation y = -x^2 + 8x + 7, follow these steps:
1. Identify the coefficient of the x-term, which is 8 in this case.
2. Take half of the coefficient and square it. Half of 8 is 4, and 4^2 is 16.
3. Add the result obtained in step 2 to both sides of the equation.
y + 16 = -x^2 + 8x + 7 + 16
y + 16 = -x^2 + 8x + 23
4. Rewrite the equation as a perfect square trinomial by factoring the quadratic expression on the right side and completing the square on the left side.
y + 16 = -(x^2 - 8x + 16) + 23
5. Simplify the equation.
y + 16 = -(x - 4)^2 + 23
6. Move the constant value to the other side of the equation.
y = -(x - 4)^2 + 7
7. Now the equation is in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Comparing this equation with the given equation, we can determine that the vertex of y = -x^2 + 8x + 7 is (4, 7).
The vertex represents the minimum or maximum point on the parabola, and the axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is x = 4.
Thus, the vertex of the parabola is (4, 7), and the axis of symmetry is x = 4.
To complete the square and determine the vertex and axis of symmetry of the equation y = -x^2 + 8x + 7, follow these steps:
Step 1: Group the x terms.
y = (-x^2 + 8x) + 7
Step 2: Take half of the coefficient of the x term and square it.
In this case, half of 8 is 4, and 4 squared is 16.
Step 3: Add the value obtained in step 2 inside the parentheses and subtract it outside the parentheses.
y = (-x^2 + 8x + 16 - 16) + 7
Step 4: Rearrange the equation.
y = (-(x^2 - 8x + 16)) + 7
Step 5: Simplify inside the parentheses by recognizing that the expression inside the parentheses is a perfect square trinomial.
y = (-(x - 4)^2) + 7
Step 6: Rearrange the equation to isolate the vertex form.
y = -(x - 4)^2 + 7
From the equation in vertex form, we can determine the vertex and axis of symmetry:
The vertex form is y = a(x-h)^2 + k, where (h, k) represents the vertex.
In this case:
a = -1, h = 4, and k = 7.
Therefore, the vertex of the parabola is (4, 7).
The axis of symmetry is given by the value of x in the vertex, so the axis of symmetry is x = 4.
To summarize:
- The vertex of the parabola is (4, 7).
- The axis of symmetry is x = 4.