Chapter 1.4, Problem 31E

To determine

To identify the parent function for the given graph, write the equation and describe the transformation for the graph.

Answer to Problem 31E

The equation is: $y=x^2-3$

The parent function is: $g(x)=x^2$

Explanation of Solution

Given:

The given graph is:

  

After looking at the graph the following can be concluded:

The given graph is a parabola.

Extrema: The graph attains its minimum value which is $y=-3$ at $x=0$ , the parabola is symmetric about y -axis. In this case $(0,-3)$ is also the y- intercept for the given graph.

Equation:

To write the equation of this parabola the standard form is used which is:

  $y=a{x}^2+bx+c$ ----------------------- (i)

Here,

a is the leading coefficient which is equal to 1 in this case.

For the parabola to be symmetric about y -axis, $b=0$

c is the y −intercept which is equal to -3 in this case.

Substituting the value of a, b and c into the equation (i)

  $y=x^2-3$

Hence, the obtained equation is: $y=x^2-3$

Verification:

Using the graphing tool the equation is plotted:

  

When the plotted graph and given graph are compared, it is found that both the graphs are equal, which means that the obtained equation is correct.

Parent function:

The obtained equation is:

  $y=x^2-3$

Therefore, the function is: $f(x)=x^2-3$

Hence, the parent function is: $g(x)=x^2$

Transformation Interpretation:

The function is: $f(x)=x^2-3$ and parent function is $g(x)=x^2$

Calculation for graph:

Consider $f(x)=x^2-3$

Values of x	Values of f (x )
0	-3
-1	-2
1	-2
2	1
-3	6

Using the above table the graph can be plotted.

Calculation for graph:

Consider $g(x)=x^2$

Values of x	Values of g (x )
1	1
-1	1
2	a
-2	4
3	9

Using the above table the graph can be plotted.

Graph:

  

Interpretation:

Hence, by comparing $f(x)$ with $g(x)$ , the following transformation can be interpreted:

The parent function is $g(x)=x^2$ and $f(x)=x^2-3$ is the rigid transformation of $g(x)$ with a vertical shift of 3 units downward.

Conclusion:

The obtained equation is $y=x^2-3$ corresponding to which the parent function is $g(x)=x^2$