Chapter 1, Problem 9REAE

Solution

Sketch the graph of the function $g\left(x\right)=\left(x+3\right)\left(x-2\right)$ and label its vertex and axis of symmetry.

Given:

The function is $g\left(x\right)=\left(x+3\right)\left(x-2\right)$ .

Concept used:

The parabola in its standard form $y=a{\left(x-h\right)}^2+k\text{ where }a\ne 0$ has vertex $\left(h,k\right)$ and axis of symmetry is the line $x=h$ .

Calculation Re-write the function $g\left(x\right)=\left(x+3\right)\left(x-2\right)$ in standard form of parabola.

  $\begin{array}{c}g\left(x\right)=\left(x+3\right)\left(x-2\right)\mathrm{......}\left(\text{Given}\right)\\ y=\left(x+3\right)\left(x-2\right)\mathrm{......}\left(\text{Substitute }g\left(x\right)=y\right)\\ y=x^2+x-6\\ y+{\left(\frac{1}{2}\right)}^2=x^2+x+{\left(\frac{1}{2}\right)}^2-6\mathrm{......}\left(\text{Add }{\left(\frac{1}{2}\right)}^2\text{ both sides}\right)\\ y+\frac{1}{4}={\left(x+\frac{1}{2}\right)}^2-6\\ y+0.25={\left(x+0.5\right)}^2-6\\ y={\left(x+0.5\right)}^2-6.25\mathrm{......}\left(\text{Standard form of parabola}\right)\end{array}$

Graph:

Construct the graph of the parabola $y={\left(x+0.5\right)}^2-6.25$ .

  

Interpretation:

The graph for the function $y={\left(x+0.5\right)}^2-6.25$ can be obtained by translating the graph of $y=x^2$ , $0.5$ units towards left and $6.25$ units downwards. And comparing $y={\left(x+0.5\right)}^2-6.25$ with the standard form of parabola, the vertex is $\left(-0.5,-6.25\right)$ and axis of symmetry is the line $x=-0.5$ as labelled in the graph.