Chapter 9.2, Problem 23PPE

To determine

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

Answer to Problem 23PPE

Equation of axis of symmetry: $x=2$

  $\text{Vertex is }\left(2,7\right).$

  $\text{y-intercept is }\left(0,3\right).$

  $\text{x-intercept is }\left(-0.6458,0\right),\left(4.6458,0\right).$

  

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=-x^2+4x+3$

Calculation:

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

  $\begin{array}{l}here;\\ a=-1\\ b=4\\ c=3\end{array}$

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

Therefore, the axis of symmetry is:

  $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{4}{2\left(-1\right)}\\ x=2\end{array}$

The x-coordinate of the given function is $x=2$ . Now, the y-coordinate will be:

  $\begin{array}{l}y=-x^2+4x+3\\ y=-{\left(2\right)}^2+4\left(2\right)+3\\ y=7\\ So,\text{Vertex is }\left(2,7\right).\end{array}$

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

  $\begin{array}{l}\text{Finding y-intercept:}\\ \text{When x = 0;}\\ \text{y = -}{\left(0\right)}^2+4\left(0\right)+3\\ y=3\\ \therefore \text{ y-intercept is }\left(0,3\right).\\ \text{Finding x-intercept:}\\ \text{When y = 0;}\\ \text{y = - }{x}^2+4x+3\\ \text{0 = - }{x}^2+4x+3\\ x=2-\sqrt{7}\left(\approx -0.6458\right),2+\sqrt{7}\left(\approx 4.6458\right)\\ \therefore \text{ x-intercept is }\left(-0.6458,0\right),\left(4.6458,0\right).\end{array}$

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

The graph will be: