Chapter 1.1, Problem 27E

Solution

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

Given information:

The function is $y=\frac{2}{3}{x}^2-3x+6$ .

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, $a=\frac{2}{3},b=-3,c=6$ .

Step-2: Find vertex $\left(x,y\right)$ where $,$ that is

  $\begin{array}{c}x=-\frac{-3}{2\cdot \left(2/3\right)}\\ =9/4\\ =2.25\\ y=f\left(-\frac{b}{2a}\right)=f\left(9/4\right)\leftarrow \text{substitute}x=9/4\text{in}y\left(x\right)\\ =\frac{2}{3}{\left(\frac{9}{4}\right)}^2-3\left(\frac{9}{4}\right)+6\\ =\frac{162}{48}-\frac{27}{4}+6\\ =\frac{162-324+288}{48}\\ =2.625\end{array}$

Thus, the vertex of parabola represented by the function v is $\left(2.25,2.625\right)$ .

Step-3: The equation of the axis of symmetry is $x=-\frac{b}{2a}$ . Here, axis of symmetry is $x=2.25$

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).

Since $a=2/3>0$ , therefore the parabola opens up.

  

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.