Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7

Answer 1

To complete the square for the quadratic function \( y = -x^2 + 8x + 7 \), we will follow these steps:

1. Factor out the coefficient of \(x^2\) (which is \(-1\)) from the first two terms: \[ y = -1(x^2 - 8x) + 7 \]

2. Complete the square for the expression inside the parentheses \(x^2 - 8x\). To do this, we take half of the coefficient of \(x\) (which is \(-8\)), square it, and then add and subtract it:

   • Half of \(-8\) is \(-4\), and squaring it gives \(16\).

   Now we add and subtract \(16\) inside the parentheses: \[ y = -1(x^2 - 8x + 16 - 16) + 7 \] Simplifying the expression inside the parentheses gives: \[ y = -1((x - 4)^2 - 16) + 7 \]

3. Distribute the \(-1\): \[ y = -1(x - 4)^2 + 16 + 7 \] \[ y = -1(x - 4)^2 + 23 \]

Now, the function is in the vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.

From the completed square form:

• The vertex \((h, k)\) is \((4, 23)\).

The axis of symmetry can be found using the \(x\)-coordinate of the vertex:

• The axis of symmetry is \(x = 4\).

Thus, the vertex is \((4, 23)\) and the axis of symmetry is \(x = 4\).

Answer 2

x=4 is the axis of symmetry of y?

Answer 3

Yes, that's correct! The equation \(x = 4\) is indeed the axis of symmetry for the quadratic function \(y = -x^2 + 8x + 7\).

In general, for a quadratic function in vertex form \(y = a(x - h)^2 + k\), the axis of symmetry is a vertical line that passes through the vertex, and it is given by the equation \(x = h\).

In this case, from the completed square form we determined that the vertex is \((4, 23)\):

• Therefore, the axis of symmetry is \(x = 4\).

This means that if you were to draw the graph of the function, it would be symmetric with respect to the line \(x = 4\).