Chapter 1.4, Problem 29E

To determine

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

Answer to Problem 29E

The equation is: $y-x=3$

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

Explanation of Solution

Given:

The given graph is:

  

After looking at the above graph the following can be concluded:

The given graph is a line.

x -intercept: The graph intersects the x -axis at about $(-3,0)$ . Hence the x -intercept is about -3.

y -intercept: The graph intersects the y -axis at about (0, 3). Hence the y -intercept is about 3.

Equation:

To write the equation of this line the intercept form which is:

  $\frac{x}{a}+\frac{y}{b}=1$ ----------------------- (i)

Where, a is the x- intercept and b is the y- intercept.

Substituting the value of intercepts into the equation (i)

  $\begin{array}{l}\frac{x}{-3}+\frac{y}{3}=1\\ \Rightarrow y-x=3\end{array}$

Hence, the obtained equation is:

  $y-x=3$ .

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:

  $y-x=3$

  $\Rightarrow y=x+3$

Consider the function: $f(x)=x+3$

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

Transformation Interpretation:

The function is: $f(x)=x+3$ and parent function is $g(x)=x$

Calculation for graph:

Consider $f(x)=x+3$

Values of x	Values of f (x )
0	3
-1	2
1	4
2	5
-2	1

Using the above table the graph can be plotted:

Calculation for graph:

Consider $g(x)=x$

Values of x	Values of g (x )
0	0
-1	-1
1	1
2	2
-2	-2

Using the above table the graph can be plotted:

Graph:

  

Interpretation:

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

The parent function is $g(x)=x$ and $f(x)$ is the rigid transformation of $g(x)$ with a horizontal shift of 3 units to the left.

Conclusion:

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