Chapter 2.6, Problem 45PPSE

Solution

To analyse the error happened in the given transformation that is supposed to transform the graph of $f\left(x\right)=x^2$ to the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$. Also, to determine the correct transformation that would transform the graph of $f\left(x\right)=x^2$ to the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$ .

Found a sequence of transformations that would give the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$ from the graph of $f\left(x\right)=x^2$ :

  $\begin{array}{l}•\text{ Shift horizontally left by 1 unit}\text{.}\\ •\text{Stretch vertically by a factor of }2.\\ •\text{Shift vertically down by }3\text{ units}\text{.}\end{array}$ .

The first two transformations of the given sequence are incorrect. The first transformation shifts the graph vertically up by $1$ unit instead of shifting it horizontally left by $1$ unit. Also, the second one shifts the graph horizontally right by $2$ units instead of making a vertical stretch by a factor of $2$ .

Given:

A transformation applied to the graph of $f\left(x\right)=x^2$ is given:

  $\begin{array}{l}•\text{ Shift vertically 1 unit up}\\ •\text{Shift horizontally 2 units right}\text{.}\\ •\text{Shift vertically 3 units down}\text{.}\end{array}$ .

Concept Used:

Horizontal Translation:

Given the graph of a function $y=f\left(x\right)$ .

The transformation $y=f\left(x-k\right)$ translates $y=f\left(x\right)$ horizontally right by $k$ units if $k>0$ .

The transformation $y=f\left(x-k\right)$ translates $y=f\left(x\right)$ horizontally left by $k$ units if $k<0$ .

Vertical Stretch:

Given the graph of a function $y=f\left(x\right)$ .

The transformation $y=af\left(x\right)$ stretches the graph of $y=f\left(x\right)$ vertically by a factor of $a$ if $a>1$ .

Vertical Translation:

Given the graph of a function $y=f\left(x\right)$ .

The transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ vertically up by $k$ units if $k>0$ .

The transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ vertically down by $k$ units if $k<0$ .

Calculation:

Find the outcome of the sequence of transformations that is given in the problem:

Consider the function $f\left(x\right)=x^2$ .

Translating the graph of $f\left(x\right)=x^2$ vertically up by $1$ unit would result in the graph of the function $h\left(x\right)$ with the equation $h\left(x\right)=x^2+1$ .

Now, shifting the graph of $h\left(x\right)=x^2+1$ horizontally right by $2$ units would result in the graph of $i\left(x\right)={\left(x-2\right)}^2+1$ .

Finally, shifting the graph of $i\left(x\right)={\left(x-2\right)}^2+1$ to the downside by $3$ units would give the graph of $j\left(x\right)={\left(x-2\right)}^2-2$ .

That is, the function $f\left(x\right)=x^2$ would have ended up in $j\left(x\right)={\left(x-2\right)}^2-2$ if it has followed the given sequence of transformations.

So, the proposed sequence of transformations is incorrect.

Find the sequence of transformations that would transform the graph of $f\left(x\right)=x^2$to the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$:

Shift the graph of $f\left(x\right)=x^2$ horizontally left by $1$ unit by applying the transformation $h\left(x\right)={\left(x+1\right)}^2$ .

Now, stretch the graph of $h\left(x\right)={\left(x+1\right)}^2$ vertically by a factor of $2$ by applying the transformation $i\left(x\right)=2{\left(x+1\right)}^2$ .

Finally, translate the graph of $i\left(x\right)=2{\left(x+1\right)}^2$ vertically down by $3$ units by applying the transformation $g\left(x\right)=2{\left(x+1\right)}^2-3$ .

Thus, the graph of $f\left(x\right)=x^2$ can be transformed to the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$ by following the sequence of transformations:

  $\begin{array}{l}•\text{ Shift horizontally left by 1 unit}\text{.}\\ •\text{Stretch vertically by a factor of }2.\\ •\text{Shift vertically down by }3\text{ units}\text{.}\end{array}$

Analyse the given transformation for errors:

The first two transformations of the given sequence are incorrect. The first transformation shifts the graph vertically up by $1$ unit instead of shifting it horizontally left by $1$ unit. Also, the second one shifts the graph horizontally right by $2$ units instead of making a vertical stretch by a factor of $2$ .

Conclusion:

Found a sequence of transformations that would give the graph of $g\left(x\right)=2{\left(x+1\right)}^2-3$ from the graph of $f\left(x\right)=x^2$ :

  $\begin{array}{l}•\text{ Shift horizontally left by 1 unit}\text{.}\\ •\text{Stretch vertically by a factor of }2.\\ •\text{Shift vertically down by }3\text{ units}\text{.}\end{array}$ .

The first two transformations of the given sequence are incorrect. The first transformation shifts the graph vertically up by $1$ unit instead of shifting it horizontally left by $1$ unit. Also, the second one shifts the graph horizontally right by $2$ units instead of making a vertical stretch by a factor of $2$ .