Chapter 1, Problem 5REAE

Solution

Sketch the graph of the function $y=x^2+2x-3$ and label its vertex and axis of symmetry.

Given:

The function is $y=x^2+2x-3$ .

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 $y=x^2+2x-3$ in its standard form.

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

Graph:

The graph for the standard parabola in standard form $y={\left(x+1\right)}^2-4$ is shown below.

  

Interpretation:

The graph for the function $y={\left(x+1\right)}^2-4$ can be obtained by translating the graph of $y=x^2$ , $1$ unit towards left and $4$ units downwards. And comparing $y={\left(x+1\right)}^2-4$ with the standard form of parabola, the vertex is $\left(-1,-4\right)$ and axis of symmetry is the line $x=-1$ as labelled in the graph.