Chapter 9.2, Problem 24PPE

To determine

The graph of each function and the axis of symmetry and vertex labelled on graph.

Answer to Problem 24PPE

Equation of axis of symmetry: $x=-\frac{5}{2}$

  $\text{Vertex is }\left(-\frac{5}{2},\frac{25}{2}\right).$

  $\text{y-intercept is }\left(0,0\right).$

  $\text{x-intercept is }\left(-5,0\right),\left(0,0\right).$

  

Explanation of Solution

Given:

The given function is:

  $y=-2x^2-10x$

Calculation:

Compare the given function with $y=a{x}^2+bx+c$ .

  $\begin{array}{l}here;\\ a=-2\\ b=-10\\ c=0\end{array}$

  $\begin{array}{l}\text{Equation of axis of symmetry is:}\\ x=-\frac{b}{2a}\end{array}$

Therefore, the axis of symmetry is:

  $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{\left(-10\right)}{2\left(-2\right)}\\ x=-\frac{5}{2}\end{array}$

The x-coordinate of the given function is $x=-\frac{5}{2}$ . Now, the y-coordinate will be:

  $\begin{array}{l}y=-2x^2-10x\\ y=-2{\left(-\frac{5}{2}\right)}^2-10\left(-\frac{5}{2}\right)\\ y=\frac{25}{2}\\ So,\text{Vertex is }\left(-\frac{5}{2},\frac{25}{2}\right).\end{array}$

Finding the other two points on the graph of the given function:

  $\begin{array}{l}\text{Finding y-intercept:}\\ \text{When x = 0;}\\ \text{y = -2}{\left(0\right)}^2-10\left(0\right)\\ y=0\\ \therefore \text{ y-intercept is }\left(0,0\right).\\ \text{Finding x-intercept:}\\ \text{When y = 0;}\\ \text{y = - 2}{x}^2-10x\\ \text{0 = - 2}{x}^2-10x\\ -2x\left(x+5\right)=0\\ x=-5,0\\ \therefore \text{ x-intercept is }\left(-5,0\right),\left(0,0\right).\end{array}$

Plot the axis of symmetry, x-intercept, y-intercept and the vertex on the graph.

The graph will be: