Chapter 8.3, Problem 8Q

To determine

To graph: The quadratic function $m\left(x\right)=-\frac{1}{2}{x}^2-4$ and compare with the graph of $f\left(x\right)=x^2$ .

Explanation of Solution

Given information:

The equation of quadratic function $m\left(x\right)=-\frac{1}{2}{x}^2-4$ .

Graph:

The graph of the quadratic function $m\left(x\right)=-\frac{1}{2}{x}^2-4$ is sketched with help of ordered pairs.

Construct a table of values with values of independent and dependent variable.

For some values of x evaluate the corresponding values of the function.

  $\begin{array}{cccccc}x& -4& -2& 0& 2& 4\\ m\left(x\right)& -12& -8& 2& -8& -12\end{array}$

Plot the ordered pairs on a coordinate plane and join the points to obtain the smooth curve.

The red curve represents the quadratic function $m\left(x\right)=-\frac{1}{2}{x}^2-4$ and the blue curve represents the function $f\left(x\right)=x^2$ .

  

Interpretation:

The quadratic functions $m\left(x\right)=-\frac{1}{2}{x}^2-4$ has vertex as $\left(0,-4\right)$ and $f\left(x\right)=x^2$ has vertex at the origin $\left(0,0\right)$ . The axis of symmetry is the y -axis. The function $f\left(x\right)=x^2$ open upward and the function $m\left(x\right)=-\frac{1}{2}{x}^2-4$ open downward. The curve $m\left(x\right)=-\frac{1}{2}{x}^2-4$ is shifted four units down from the function $f\left(x\right)=x^2$ on the y- axis as it is the vertical translation of the function $f\left(x\right)=x^2$ by $\frac{1}{2}$ units downward.