Chapter SH, Problem 9.2EP

To determine

To find: The vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $\mathrm{y}=-2{\mathrm{x}}^2+8\mathrm{x}+5$ .

Answer to Problem 9.2EP

The vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $\mathrm{y}=-2{\mathrm{x}}^2+8\mathrm{x}+5$ are $\left(2,13\right)$ , $\mathrm{x}=2$ and 5, respectively.

Explanation of Solution

Given information:

The given equation is $\mathrm{y}=-2{\mathrm{x}}^2+8\mathrm{x}+5$ .

Calculation:

The given equation is a quadratic equation.

  $\mathrm{y}=\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}$

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

  $\mathrm{a}=-2,\mathrm{b}=8,\mathrm{c}=5$

Write the equation for the axis of the symmetry.

  $\mathrm{x}=\frac{-\mathrm{b}}{2\mathrm{a}}$

Substitute $8$ for $\mathrm{b}$ and $-2$ for $\mathrm{a}$ .

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

So, the equation for the axis of the symmetry is $\mathrm{x}=2$ .

The vertex is equal to $\left(\mathrm{x},\mathrm{y}\right)$ . To calculate the value of $\mathrm{y}$ , substitute 2 for $\mathrm{x}$ in the given equation.

  $\begin{array}{c}\mathrm{y}=-2{\left(2\right)}^2+8\times 2+5\\ =-8+16+5\\ =13\end{array}$

So, the vertex is $\left(2,13\right)$ .

The y -intercept of the graph of the equation is equal to the value of $\mathrm{c}$ . So, the y -intercept of the graph of the equation is 5.

Therefore, the vertex, the equation of the axis of symmetry, and the y -intercept of the graph of the equation $\mathrm{y}=-2{\mathrm{x}}^2+8\mathrm{x}+5$ are $\left(2,13\right)$ , $\mathrm{x}=2$ and 5, respectively.