Chapter 1.1, Problem 6GP

Solution

To graph: The function and label the vertex and axis of symmetry.

Given information:

The function is $y\left(x\right)=-\frac{1}{3}{x}^2-5x+2$ .

Graph:

The steps followed for graphing the function of the form $y=a{x}^2+bx+c$ are:

Step-1: Find coefficients $a,b,c$ . Here, $a=-\frac{1}{3},b=-5,c=2$ .

Step-2: Find vertex $\left(x,y\right)$ where $x=-\frac{b}{2a},y=y\left(-\frac{b}{2a}\right)$ that is $\begin{array}{c}x=-\frac{-5}{2\cdot \left(-1/3\right)}\\ =-7.5\\ y=y\left(-7.5\right)\leftarrow \text{substitute}x=1\text{in}y\left(x\right)\\ =-\frac{1}{3}{\left(-7.5\right)}^2-5\left(-7.5\right)+2\\ =-18.75+37.5+2\\ =20.75\end{array}$

Thus, the vertex of the parabola represented by the function $y\left(x\right)=2x^2+6x+3$ is $\left(-7.5,20.75\right)$ .

Step-3: The equation of the axis of symmetry (line) is $x=-\frac{b}{2a}$ . Here, the axis of symmetry is the line $x=-7.5$ .

Step-4: Find other points $\left(a,y\left(a\right)\right)$ on the curve.

Step-5: Plot axis of symmetry, vertex and other points. The points table and the graph of the function is shown in Fig. (1).

  

Interpretation:

The axis of symmetry divides the area of the region bounded by the curve into two equal parts.

The vertex of the graph of a function is a point where curve bends making this point a lowest or highest point in the curve where maxima or minima of the function exists.