Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
To complete the square, let's begin by re-writing the equation:
y = -x^2 + 8x + 7
First, let's take out the constant term (+7) from the quadratic part (-x^2 + 8x), so we have:
y = -(x^2 - 8x) + 7
Now, let's complete the square. We take half of the x-coefficient (which is 8) and square it:
(-8/2)^2 = (-4)^2 = 16
Now, we add and subtract 16 inside the parentheses:
y = -(x^2 - 8x + 16 - 16) + 7
Simplifying further:
y = -(x^2 - 8x + 16) + 16 + 7
y = -(x - 4)^2 + 23
Now, we can see that the equation has been written in the form y = a(x - h)^2 + k, where h and k are the x and y coordinates of the vertex, respectively.
In this case, the vertex is (h, k) = (4, 23), so the x-coordinate of the vertex is 4 and the y-coordinate is 23.
To find the axis of symmetry, which is a vertical line through the vertex, the equation is simply 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:
1. First, make sure that the coefficient of the x^2 term is 1. Since it is already -1 in this equation, we do not need to make any changes.
2. To complete the square, we need to focus on the x term (8x in this case). Take half of the coefficient of the x term and square it. Half of 8 is 4, and 4 squared is 16. Add this value to both sides of the equation.
y - 16 = -x^2 + 8x + 7 + 16
Simplify the right side:
y - 16 = -x^2 + 8x + 23
3. Now, rewrite the right side of the equation as a perfect square trinomial. To do this, write (x + a)^2, where 'a' is half of the coefficient of the x term (which is 4).
y - 16 = -(x^2 - 8x + 16) + 23
Simplify the right side further:
y - 16 = -(x - 4)^2 + 23
4. Finally, isolate y by adding 16 to both sides of the equation:
y = -(x - 4)^2 + 39
Now we have the equation in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
From this equation, we can determine the vertex and axis of symmetry:
- The vertex is located at the point (h, k). In this case, the vertex is (4, 39).
- The axis of symmetry is a vertical line passing through the vertex, so the equation of the axis of symmetry is x = h. In this case, 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: Separate the constant term from the x terms:
y = (-x^2 + 8x) + 7
Step 2: Factor out the coefficient of x^2 from the x terms:
y = -(x^2 - 8x) + 7
Step 3: To complete the square, take half of the coefficient of x and square it. Add this value inside the parentheses and subtract it outside of the parentheses to maintain the equation's balance:
y = -(x^2 - 8x + (8/2)^2) + 7 - (8/2)^2
y = -(x^2 - 8x + 16) + 7 - 16
Step 4: Simplify inside the parentheses and combine constants:
y = -(x - 4)^2 - 9
Step 5: Rearrange the equation in vertex form:
y = -(x - 4)^2 - 9
Comparing this equation to the standard vertex form, y = a(x - h)^2 + k, we can identify the vertex (h, k).
The vertex of this equation is (h, k) = (4, -9).
The axis of symmetry is a vertical line passing through the x-coordinate of the vertex. Therefore, the axis of symmetry is x = 4.