Chapter 3.3, Problem 31AYU

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 $f\left(x\right)=a{(x-h)}^2+k$.]

$f\left(x\right)=x^2+x-1$

To determine

To graph: The transformation of the function $f(x)=\frac{1}{2}{x}^2+x-1$.

Answer to Problem 31AYU

Explanation of Solution

Given:

We have to graph the function $f(x)=\frac{1}{2}{x}^2+x-1$.

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 $f(x)=\frac{1}{2}{x}^2+x-1$.

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 $f(x)=\frac{1}{2}(x^2+2x)-1$

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

Therefore, we get

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

Thus, the given equation can be written in the standard form as $f(x)=\frac{1}{2}{\left(x+1\right)}^2-\frac{3}{2}$.

Since, $a=\frac{1}{2}$ is positive, the graph opens upwards.

We can see that $a=\frac{1}{2}$ is lesser, therefore the graph is short and wide.

Here, the value of $f(x)=\frac{1}{2}{x}^2$ is half the value of the function $f(x)=x^2$.

In the given function, we have $h=-1$ and $k=-\frac{3}{2}$, therefore, the graph is the graph of $f(x)=x^2$ opening upwards with 1 unit shifted horizontally left and 1.5 units shifted vertically downwards.