Chapter SH, Problem 9.3EP

To determine

To find: The vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $y=x^2-8x+9$ .

Answer to Problem 9.3EP

The vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $y=x^2-8x+9$ are $\left(4,-7\right)$ , $x=4$ and 9, respectively.

Explanation of Solution

Given information:

The given equation is $y=-2x^2+8x+5$ .

Calculation:

The given equation is a quadratic equation.

  $y=a{x}^2+bx+c$

Compare the given equation with the standard form of the quadratic equation.

  $a=1,b=-8,c=9$

Write the equation for the axis of the symmetry.

  $x=\frac{-b}{2a}$

Substitute $-8$ for $b$ and $1$ for $a$ .

  $\begin{array}{c}x=\frac{-\left(-8\right)}{2\left(1\right)}\\ =\frac{8}{2}\\ =4\end{array}$

So, the equation for the axis of the symmetry is $x=4$ .

The vertex is equal to $\left(x,y\right)$ . To calculate the value of $y$ , substitute 4 for $x$ in the given equation.

  $\begin{array}{c}y={\left(4\right)}^2-8\times 4+9\\ =16-32+9\\ =-16+9\\ =-7\end{array}$

So, the vertex is $\left(4,-7\right)$ .

The y -intercept of the graph of the equation is equal to the value of $c$ . So, the y -intercept of the graph of the equation is 9.

Therefore, the vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $y=x^2-8x+9$ are $\left(4,-7\right)$ , $x=4$ and 9, respectively.