Chapter 1, Problem 7REAE

Solution

Sketch the graph of the function $f\left(x\right)=-x^2-2x-6$ and label its vertex and axis of symmetry.

Given:

The function is $f\left(x\right)=-x^2-2x-6$ .

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 $f\left(x\right)=-x^2-2x-6$ in its standard form.

  $\begin{array}{c}f\left(x\right)=-x^2-2x-6\mathrm{......}\left(\text{Given}\right)\\ y=-x^2-2x-6\mathrm{......}\left(\text{Write }f\left(x\right)=y\right)\\ y=-\left(x^2+2x\right)-6\\ y-1=-\left(x^2+2x+1\right)-6\mathrm{......}\left(\text{Complete square}\right)\\ y-1=-{\left(x+1\right)}^2-6\\ y=-{\left(x+1\right)}^2-5\mathrm{......}\left(\text{Standard form of parabola}\right)\end{array}$

Graph:

The graph for the function $y=-{\left(x+1\right)}^2-5$ is shown below.

  

Interpretation:

From the graph it can be interpreted that the graph of $y=-{\left(x+1\right)}^2-5$ is a downward parabola which can be obtained by translating the graph of $y=x^2$ , $1$ unit towards left and $5$ units downwards. And comparing $y=-{\left(x+1\right)}^2-5$ with the standard form of parabola, the vertex is $\left(-1,-5\right)$ and axis of symmetry is the line $x=-1$ as labelled in the graph.