Chapter 8, Problem 2CT

To determine

To draw: the Graph for given function and compare the given graph to a parent function.

Explanation of Solution

Given information:

Parent function: $f\left(x\right)=x^2$

Transformed function: $g\left(x\right)=-\frac{1}{2}{x}^2$

Graphing the function: $g\left(x\right)=-\frac{1}{2}{x}^2$

Comparing this equation with the vertex form of quadratic equation:

  $y=a{\left(x-h\right)}^2+k$

  $g\left(x\right)=-\frac{1}{2}{x}^2$

On comparing we find that, the vertex ( h, k ) is $\left(0,0\right)$ and $a=-\frac{1}{2}$ .

As the value of a is a negative number, so the graph will open down.

Now, take some random points of x and y to plot more points on graph:

x	y
-2	-2
2	-2

Now plot these points on the graph:

Therefore, the graph is

  

Now, let us make the graph of $f\left(x\right)=x^2$ .

Comparing this equation with the vertex form of quadratic:

  $y=a{\left(x-h\right)}^2+k$

  $f\left(x\right)=x^2$

On comparing we find that the vertex of equation ( h, k ) is $\left(0,0\right)$ and $a=1$ .

As the value of a is a positive number, so the graph will open up.

Now take more points to plot on the graph:

x	y
2	4
-2	4

Now plot these points on the graph:

  

Let us compare the graph of $f\left(x\right)=x^2$ and graph of $\text{g}\left(x\right)=-\frac{1}{2}{x}^2$ .

We could see $\text{ the graph of g}\left(x\right)\text{ is reflected accross }x-axis$ .

Also, if we compare y -coordinate value of point (-2,4) by y -coordinate value of point (-2,-2),

  $\text{the graph of g(}x)\text{ compress vertically by a factor of }\frac{1}{2}.$

Therefore, the graph of g( x) is reflected across x -axis and compress vertically by a factor of $\frac{1}{2}$ of function $f\left(x\right).$