Chapter 3, Problem 1RE

(a)

To determine

To graph: The quadratic function $f\left(x\right)=2x^2$ and compare with the graph of $y=x^2$.

Explanation of Solution

Graph:

Consider the general quadratic function in standard form.

$f\left(x\right)=a{\left(x-h\right)}^2+k,a\ne 0$

Here, $\left|a\right|$ decides the width of the parabola. If $\left|a\right|<1$, then the parabola shrinks vertically and if $\left|a\right|>1$, then the graph stretches vertically. Also, if $a<0$, then the parabola has a reflection in the $x\text{-axis}$.

Now, consider the function,

$f\left(x\right)=2x^2$

Here, $a=2$ which is greater than 1, therefore the graph of the function $f\left(x\right)=2x^2$ is obtained by shrinking the graph of the parabola $y=x^2$ vertically by the factor of $2$ which produces a narrower parabola.

Then, the graph for the function $f\left(x\right)=2x^2$ and $y=x^2$ can be drawn as,

(b)

To determine

To graph: The quadratic function $g\left(x\right)=-2x^2$ and compare with the graph of $y=x^2$.

Explanation of Solution

Graph:

Consider the function,

$g\left(x\right)=-2x^2$

Here, $a=-2$, but $\left|a\right|=2$ which is greater than 1, therefore the graph of the function $g\left(x\right)=-2x^2$ is obtained by shrinking the graph of the parabola $y=x^2$ vertically by the factor of $2$ which produces a narrower parabola. Also, $a<0$, therefore the parabola has a reflection in the $x\text{-axis}$.

Then, the graph for the function $g\left(x\right)=-2x^2$ from $y=x^2$ can be drawn as,

(c)

To determine

To graph: The quadratic function $h\left(x\right)=x^2+2$ and compare with the graph of $y=x^2$.

Explanation of Solution

Graph:

Now, consider the function,

$h\left(x\right)=x^2+2$

Here, $a=1$ and $k=2$, therefore the graph of the function $h\left(x\right)=x^2+2$ is obtained by shifting the graph of $y=x^2$ vertically upward by 2 units.

Then, the graph for the function $h\left(x\right)=x^2+2$ from $y=x^2$ can be drawn as,

(d)

To determine

To graph: The quadratic function $k\left(x\right)={\left(x+2\right)}^2$ and compare with the graph of $y=x^2$.

Explanation of Solution

Graph:

Now, consider the function,

$k\left(x\right)={\left(x+2\right)}^2$

Here, $a=1$ and $h=-2$, therefore the graph of the function $k\left(x\right)={\left(x+2\right)}^2$ is obtained by shifting the graph of the parabola $y=x^2$ horizontally by the factor of $2$ towards left.

Then, the graph for the function $k\left(x\right)={\left(x+2\right)}^2$ from $y=x^2$ can be drawn as,