Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7
To complete the square in the quadratic equation y = -x^2 + 8x + 7, we need to focus on the terms involving x.
First, let's rearrange the equation:
y = -(x^2 - 8x) + 7
Next, we want to take half of the coefficient of x, square it, and add it to both sides:
y = -(x^2 - 8x + 16) + 7 + 16
= -(x^2 - 8x + 16) + 23
Now, we can rewrite the expression inside the parentheses as a perfect square:
y = -(x^2 - 8x + 16) + 23
= -(x - 4)^2 + 23
The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex.
From our equation, we can see that the vertex is (4, 23), and the axis of symmetry is x = 4.
To complete the square and determine the vertex and axis of symmetry for the quadratic equation y = -x^2 + 8x + 7, follow these steps:
Step 1: Identify the coefficients.
The equation is already in standard form, y = ax^2 + bx + c, where a = -1, b = 8, and c = 7.
Step 2: Find the constant term to complete the square.
To complete the square, divide the coefficient of 'b' by 2, square the result, and add it to both sides of the equation.
In this case, b/2 = 8/2 = 4, and (b/2)^2 = 4^2 = 16.
Adding 16 to both sides of the equation gives us:
y + 16 = -x^2 + 8x + 7 + 16
y + 16 = -x^2 + 8x + 23
Step 3: Write as a perfect square trinomial.
Rearrange the terms on the right side of the equation to write it as a perfect square trinomial.
y + 16 = (-(x^2 - 8x)) + 23
y + 16 = (-(x^2 - 8x + 16)) + 23
To maintain equality, subtract 16 inside the parentheses, but also subtract 16 outside to balance the equation:
y + 16 - 16 = -(x^2 - 8x + 16) + 23 - 16
y = -(x^2 - 8x + 7)
Now, the right side is a perfect square trinomial.
Step 4: Rewrite as a binomial squared.
Write the perfect square trinomial as a binomial squared, and simplify the equation.
y = -((x - 4)^2 - 7)
y = -(x - 4)^2 + 7
Step 5: Determine the vertex and axis of symmetry.
The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. In this case, within the rewritten equation, the vertex is (4, 7).
The axis of symmetry is a vertical line passing through the vertex. In this case, it is the line x = 4.
Therefore, the vertex of the parabola is (4, 7), and the axis of symmetry is x = 4.
To complete the square, we need to rewrite the equation in the form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Given the equation y = -x^2 + 8x + 7, let's start by rearranging the terms:
y = -(x^2 - 8x) + 7
Now, we need to create a perfect square trinomial by adding and subtracting a constant value inside the parentheses. We'll complete the square with respect to x - 4, since we want the coefficient of the x-term (here, 8x) to be twice the coefficient of the x-term in (x - 4)^2.
y = -(x^2 - 8x + 16 - 16) + 7
Inside the parentheses, we have a perfect square trinomial (x - 4)^2 - 16. Simplifying further:
y = -(x - 4)^2 + 16 - 7
y = -(x - 4)^2 + 9
Now, we have our equation in the desired form, where a = -1, h = 4, and k = 9.
The vertex is given by (h, k), so the vertex is (4, 9).
The axis of symmetry is the vertical line passing through the vertex, so the equation of the axis of symmetry is x = 4.