Chapter 3.3, Problem 30AYU

In Problems 21-32, graph the function $f$ by starting with the graph of $y=x^2$ and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

[ Hint: If necessary, write $f$ in the form $f\left(x\right)=a{(x-h)}^2+k$.]

$f\left(x\right)=-2x^2+6x+2$

To determine

To graph: The transformation of the function $f(x)=-2x^2+6x+2$.

Answer to Problem 30AYU

Explanation of Solution

Given:

We have to graph the function $f(x)=-2x^2+6x+2$.

Graph:

The graph of the given function is

Interpretation:

The graph of $f(x)=a{x}^2,a=1$ is

Now, we can graph the given function by transforming the above graph.

The given function is $f(x)=-2x^2+6x+2$.

The general form of a quadratic function is $g(x)=a{\left(x-h\right)}^2+k$.

Here, if $a<0$, the graph opens upward otherwise the graph opens downwards.

If $a$ is closer to 0, then the graph is shorter and wider.

If $a$ is large, then the graph is tall and narrow.

Then, the graph of $g(x)$ is the graph of $f(x)$ with $h$ units shifted horizontally and $k$ units shifted vertically.

Here, we can convert the given equation into the standard form by using the square completion technique.

The given equation can also be written as $f(x)=-2(x^2-3x)+2$

Therefore, let us now add and subtract the square of half of the $x\text{-coefficient}$ in the given equation.

Therefore, we get

$\begin{array}{l}f(x)=-2(x^2-3x)+2\\ \Rightarrow -2\left(x^2-3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2\right)+2\\ \Rightarrow -2\left({\left(x-\frac{3}{2}\right)}^2-\left(\frac{9}{4}\right)\right)+2\\ \Rightarrow -2\left({\left(x-\frac{3}{2}\right)}^2\right)+\left(\frac{9}{2}\right)+2\\ \Rightarrow -2\left({\left(x-\frac{3}{2}\right)}^2\right)+\left(\frac{13}{2}\right)\end{array}$

Thus, the given equation can be written in the standard form as $f(x)=-2\left({\left(x-\frac{3}{2}\right)}^2\right)+\left(\frac{13}{2}\right)$.

Since, $a=-2$ is negative, the graph opens downwards.

We can see that $\left|a\right|=2$ is larger, therefore the graph is tall and narrow.

Here, the value of $f(x)=-2x^2$ is twice the value of the function $f(x)=x^2$.

In the given function, we have $h=\frac{3}{2}$ and $k=\frac{13}{2}$, therefore, the graph is the graph of $f(x)=x^2$ opening downwards with 1.5 units shifted horizontally right and 6.5 unit shifted vertically upwards.