Chapter 1, Problem 3T

Solution

Sketch the graph of the function $f\left(x\right)=2\left(x+4\right)\left(x-2\right)$ and label its vertex and axis of symmetry.

Given:

The function is $f\left(x\right)=2\left(x+4\right)\left(x-2\right)$ .

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 $x=h$ . And construct the graph by translating the graph of $y=x^2$ .

Calculation Re-write the function $f\left(x\right)=2\left(x+4\right)\left(x-2\right)$ in standard form of parabola.

  $\begin{array}{c}f\left(x\right)=2\left(x+4\right)\left(x-2\right)\mathrm{......}\left(\text{Given}\right)\\ y=2\left(x+4\right)\left(x-2\right)\mathrm{......}\left(\text{Substitute }f\left(x\right)=y\right)\\ y=2\left(x^2+2x-8\right)\\ y=2\left(x^2+2x\right)-16\\ y+2=2\left(x^2+2x+1\right)-16\mathrm{......}\left(\text{Add 2 both sides}\right)\\ y+2=2{\left(x+1\right)}^2-16\\ y=2{\left(x+1\right)}^2-18\mathrm{......}\left(\text{Standard form of parabola}\right)\end{array}$

Graph:

The graph for the function $y=2{\left(x+1\right)}^2-18$ in standard form of parabola is shown below.

  

Interpretation:

The graph for the function $y=2{\left(x+1\right)}^2-18$ can be obtained by translating the graph of $y=x^2$ , $1$ unit towards left and $18$ units downwards. And comparing $y=2{\left(x+1\right)}^2-18$ with the standard form of parabola, the vertex is $\left(-1,-18\right)$ and axis of symmetry is the line $x=-1$ as labelled in the graph.