Chapter 1.4, Problem 32E

To determine

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

Answer to Problem 32E

The equation is: $y=|x+2|$

The parent function is: $g(x)=\left|x\right|$

Explanation of Solution

Given:

The given graph is:

  

After looking at the graph the following can be concluded:

The given graph is transformation of $y=\left|x\right|$ with a horizontal shift.

x -intercept: The graph intersects the x -axis at about $(-2,0)$ ,hence the x -intercept is about -2.

y- intercept: The graph intersects the y- axis at about (0,2), hence the x- intercept is about 2.

Equation:

To write the equation of the given graph the standard form is used which is:

  $y=|x-a|$ ----------------------- (i)

Here,

a is x- intercept of the given graph.

Substituting the value of a into the equation (i),

  $y=|x+2|$

Hence, the obtained equation is: $y=|x+2|$

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:

  $y=|x+2|$

Therefore, the function is: $f(x)=|x+2|$

Hence, the parent function is: $g(x)=\left|x\right|$

Transformation Interpretation:

The function is $f(x)=|x+2|$ and parent function is $g(x)=\left|x\right|$

Calculation for graph:

Consider $f(x)=|x+2|$

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

Using the above table the graph can be plotted.

Calculation for graph:

Consider $g(x)=\left|x\right|$

Values of x	Values of g (x )
1	1
-1	1
2	2
-2	2
3	3

Using the above table the graph can be plotted.

Graph:

  

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

The parent function is $g(x)=\left|x\right|$ and $f(x)=|x+2|$ is the rigid transformation of $g(x)$ with a horizontal shift of 2 units towards left.

Conclusion:

The obtained equation is $y=|x+2|$ corresponding to which the parent function is $g(x)=\left|x\right|$