Chapter 4.3, Problem 49E

To determine

To find: the transformation that have take place between the parent graph and the another grpah.

Answer to Problem 49E

The graph of $y=0.5{\left(x+1\right)}^2$ is obtained by translating the graph $y=x^2$ to the left one unit and compressing it vertically.

Explanation of Solution

Given:

  $y=x^2\text{ and }y=0.5{\left(x+1\right)}^2.$

Concept used:

To move the function up add outside the function as:

  $f\left(x\right)+b$ is $f\left(x\right)$ moved up by $b$ units.

Moving the function down works the same way as $f\left(x\right)-b$ is $f\left(x\right)$ move down $b$ units.

To shift the function left, add inside the function’s argument:

  $f\left(x+b\right)$ gives $f\left(x\right)$ shifted $b$ units to the left.

Shifting to the right work the same way as $f\left(x-b\right)$ is $f\left(x\right)$ shifted $b$ units to the rights.

A function $f\left(x\right)$ , a new function $g\left(x\right)=kf\left(x\right)$ , where $a$ is the constant, is a vertical stretch or vertical compression of the function $f\left(x\right)$ .

If $a>1$ then graph will be stretched.

If $0<a<1$ , then the graph will be compressed.

If $a<0$ ,then there will be combination of a vertical stretch or compression with a vertical reflection.

Or

  $g\left(x\right)=f\left(kx\right)$

The graph can be sketch by horizontally shrinking, $f\left(x\right)$ by the factor of $\frac{1}{k}$ .

Calculation:

Consider the equations:

  $y=x^2\text{ and }y=0.5{\left(x+1\right)}^2.$

To graph the equation $y=0.5{\left(x+1\right)}^2$ , consider the graph of parent function $y=x^2$ .

The graph of $y=0.5{\left(x+1\right)}^2$ is obtained by translating the graph $y=x^2$ to the left one unit and compressing it vertically.

The parent graph $y=x^2$ and the given graph $y=0.5{\left(x+1\right)}^2$ are shown below.

  

Hence, the graph of $y=0.5{\left(x+1\right)}^2$ is obtained by translating the graph $y=x^2$ to the left one unit and compressing it vertically.