Chapter 1.5, Problem 59E

To determine

(a)

To draw:

The scatterplot of the given data.

To determine

(b)

The quadratic function that approximate the given data.

Solution:

The quadratic function that approximate the given data is $y=0.068x^2+8.73x+400.63$.

Conclusion:

Explanation:

Approach:

The procedure to obtain the quadratic function is,

• Press $\boxed{\text{STAT}}$ button then select $Edit$ option.
• Enter the points to be plotted.
• Press $\boxed{\text{STAT}}$ button and select $Calc$ option.
• Now select option $5$ $\boxed{\text{Quad}\text{Reg}}$ and hit $Enter$ twice.
• Press $\boxed{\text{Y=}}$ button, enter the quadratic equation and select plot $1$.
• Press the $\boxed{\text{Graph}}$ button.
• Press $\boxed{\text{Window}}$ button and set the scale.

Calculation:

Press $\boxed{\text{STAT}}$ button then select $Edit$ option.

Enter the points to be plotted.

The resulting screen is shown in figure $\left(1\right)$.

Press $\boxed{\text{STAT}}$ button and select $Calc$ option.

Now select option $5$ $\boxed{\text{Quad}\text{Reg}}$ and hit $Enter$ twice.

The resulting screen is shown below.

Figure $\left(3\right)$

The standard form of a quadratic equation is given as,

$y=a{x}^2+bx+c$

It is obtained from figure $\left(3\right)$.

$\begin{array}{l}a=0.068\\ b=8.73\\ c=400.63\end{array}$

The quadratic function obtained is,

$y=0.068x^2+8.73x+400.63$

Therefore, the quadratic function that approximate the given data is $y=0.068x^2+8.73x+400.63$.

Conclusion:

Hence, the quadratic function that approximate the given data is $y=0.068x^2+8.73x+400.63$.

To determine

(c)

To graph:

The quadratic function on the same window as scatter plot

Solution:

The graph of the quadratic function on the same window as scatter plot is shown in figure $\left(5\right)$.

Explanation:

Approach:

The procedure to obtain the quadratic function is,

• Press $\boxed{\text{STAT}}$ button then select $Edit$ option.
• Enter the points to be plotted.
• Press $\boxed{\text{STAT}}$ button and select $Calc$ option.
• Now select option $5$ $\boxed{\text{Quad}\text{Reg}}$ and hit $Enter$ twice.
• Press $\boxed{\text{Y=}}$ button, enter the quadratic equation and select plot $1$.
• Press the $\boxed{\text{Graph}}$ button.
• Press $\boxed{\text{Window}}$ button and set the scale.

Calculation:

The quadratic function obtained is,

$y=0.068x^2+8.73x+400.63$

Press $\boxed{\text{Y=}}$ button, enter the quadratic equation and select plot $1$.

The resulting screen is shown below.

Figure $\left(4\right)$

Press the $\boxed{\text{Graph}}$ button.

The resulting screen is shown below.

Figure $\left(5\right)$

Therefore, the graph of the quadratic function on the same window as scatter plot is shown in figure $\left(5\right)$.

Conclusion:

Hence, the graph of the quadratic function on the same window as scatter plot is shown in figure $\left(5\right)$.

To determine

(d)

The cubic function that approximate the given data.

To determine

(e)

To graph:

The quadratic function on the same window as scatter plot.

Solution:

The graph of the quadratic function on the same window as scatter plot is shown in figure $\left(5\right)$.

Explanation:

Approach:

The procedure to obtain the quadratic function is,

• Press $\boxed{\text{STAT}}$ button then select $Edit$ option.
• Enter the points to be plotted.
• Press $\boxed{\text{STAT}}$ button and select $Calc$ option.
• Now select option $5$ $\boxed{\text{Quad}\text{Reg}}$ and hit $Enter$ twice.
• Press $\boxed{\text{Y=}}$ button, enter the quadratic equation and select plot $1$.
• Press the $\boxed{\text{Graph}}$ button.
• Press $\boxed{\text{Window}}$ button and set the scale.

Calculation:

The quadratic function obtained is,

$y=-0.0133x^3+1.285x^2-19.61x+491.5$

Press $\boxed{\text{Y=}}$ button, enter the quadratic equation and select plot $1$

The resulting screen is shown below.

Figure $\left(7\right)$

Press the $\boxed{\text{Graph}}$ button.

The resulting screen is shown below.

Figure $\left(8\right)$

Therefore, the graph of the cubic function on the same window as scatter plot is shown in figure $\left(8\right)$.

Conclusion:

Hence, the graph of the cubic function on the same window as scatter plot is shown in figure $\left(8\right)$.

To determine

(f)

The function that fits best for the data with reason.