Chapter 2.1, Problem 47E

a.

To determine

To identify: The values of $a,h,\text{ and }k$ and use vertex from to write the transformed function.

Answer to Problem 47E

The obtained equation is $g\left(x\right)=2{\left(x-1\right)}^2+6$ .

The values are $a=2,h=1,k=6$

Explanation of Solution

Given information: The graph of $g$ is a translation of the graph $f\left(x\right)=x^2$ ,3 units upwards and 1 unit right followed by a vertical stretch by a factor of 2.

Concept Used:

The equation of the form $y=a{\left(x-h\right)}^2+k$ represents a parabola with vertex $\left(h,k\right)$ .

Calculation:

Consider the graph $f\left(x\right)=x^2$ and translate the graph 3 units upwards,

  $\begin{array}{l}f\left(x\right)=x^2\\ f\left(x\right)+3=x^2+3\end{array}$

Now, translate the graph by 1 unit right,

  $\begin{array}{c}f\left(x\right)=x^2\\ f\left(x\right)+3=x^2+3\\ f\left(x-1\right)+3={\left(x-1\right)}^2+3\end{array}$

Further, stretch the graph vertically by a factor of 2.

  $\begin{array}{c}f\left(x\right)=x^2\\ f\left(x\right)+3=x^2+3\\ f\left(x-1\right)+3={\left(x-1\right)}^2+3\\ 2\left[f\left(x-1\right)+3\right]=2\left[{\left(x-1\right)}^2+3\right]\\ g\left(x\right)=2\left[{\left(x-1\right)}^2+3\right]\\ g\left(x\right)=2{\left(x-1\right)}^2+6\mathrm{......}\left(1\right)\end{array}$

Compare equation (1) with $y=a{\left(x-h\right)}^2+k$ and write the values of $a,h,\text{ and }k$ .

  $\begin{array}{l}a=2\\ h=1\\ k=6\end{array}$

b.

To determine

To write: The transformed function and compare this function with the function in part (a) using function notation

Answer to Problem 47E

The function is same as the function obtained in part a.

Explanation of Solution

Given: The graph of $g$ is a translation of the graph $f\left(x\right)=x^2$ ,3 units upwards and 1 unit right followed by a vertical stretch by a factor of 2.

Concept Used:

The equation of the form $y=a{\left(x-h\right)}^2+k$ represents a parabola with vertex $\left(h,k\right)$ .

Calculation:

  $\begin{array}{c}f\left(x\right)=x^2\\ \to x^2+3\\ \to {\left(x-1\right)}^2+3\\ \to 2\left[{\left(x-1\right)}^2+3\right]\\ \to 2\left[{\left(x-1\right)}^2+3\right]\\ \to 2{\left(x-1\right)}^2+6\mathrm{......}\left(2\right)\end{array}$

c.

To determine

To Identify: $a,h,\text{ and }k$ and use vertex form to write the transformed function and compare the equation obtained by using the function notation to write the transformed equation.

Answer to Problem 47E

The obtained equation is $g\left(x\right)=2{\left(x-1\right)}^2+3$ .

The values are $a=2,h=1,k=3$

The obtained equation is different when different translations are carried out.

Explanation of Solution

Given information: The graph of $g$ is a translation of the graph $f\left(x\right)=x^2$ ,3 units upwards and 1 unit right followed by a vertical stretch by a factor of 2.

Concept Used:

The equation of the form $y=a{\left(x-h\right)}^2+k$ represents a parabola with vertex $\left(h,k\right)$ .

Calculation:

Consider the graph $f\left(x\right)=x^2$ and perform a vertical shift of 2 by the graph.

  $\begin{array}{l}f\left(x\right)=x^2\\ 2f\left(x\right)=2x^2\end{array}$

Now, translate the graph by 3 units upward and 1 unit right,

  $\begin{array}{c}f\left(x\right)=x^2\\ 2f\left(x\right)=2x^2\\ 2f\left(x-1\right)+3=2{\left(x-1\right)}^2+3\end{array}$

Compare equation (1) with $y=a{\left(x-h\right)}^2+k$ and write the values of $a,h,\text{ and }k$ .

  $\begin{array}{l}a=2\\ h=1\\ k=3\end{array}$

Write the equation using function notations,

  $\begin{array}{c}f\left(x\right)=x^2\\ \to 2x^2\\ \to 2{\left(x-1\right)}^2+3\mathrm{......}\left(3\right)\end{array}$

c.

To determine

The preference of function to translate the equations.

Answer to Problem 47E

I prefer function notation method.

Explanation of Solution

Given: Two methods of translating the function.

Calculation:

I prefer function notation to do the translations as it clearly mentions step by step translation using an arrow.