Answer :
Sure, let's go through the solution step-by-step. We will sketch the graph of the function [tex]\( f(x) = -0.589 x^2 + 38.1 x - 561 \)[/tex] and plot the given mileage data point.
### Step-by-Step Solution:
1. Understand the Function:
The function given is [tex]\( f(x) = -0.589 x^2 + 38.1 x - 561 \)[/tex]. This formulates a parabola. Since the coefficient of [tex]\( x^2 \)[/tex] is negative, the parabola opens downwards.
2. Mileage Data Provided:
The data provided in the table is for [tex]\( x = 28 \)[/tex] pounds per square inch, and the mileage is 43 thousand miles.
3. Plotting the Function:
To sketch the graph, we need to plot the function over a range of [tex]\( x \)[/tex]-values. Typically, [tex]\( x \)[/tex]-values of tire pressure can range from 0 to 50 pounds per square inch.
4. Calculate [tex]\( f(x) \)[/tex] Values for a Range of [tex]\( x \)[/tex]:
We need to evaluate the function [tex]\( f(x) \)[/tex] at various [tex]\( x \)[/tex]-values within our chosen range to get the shape of the parabola.
5. Plot the Function and the Data Point:
- For each [tex]\( x \)[/tex]-value in our range, compute [tex]\( f(x) \)[/tex].
- Plot these points to visualize the function.
- Plot the specific data point given: [tex]\( (28, 43) \)[/tex].
### Creating a Table of Values:
Let's create a table of values for [tex]\( x \)[/tex] from 0 to 50 with an interval of 5, and calculate the corresponding [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & -561 \\
5 & 132.825 \\
10 & 557 \\
15 & 711.525 \\
20 & 596.4 \\
25 & 211.625 \\
30 & -441.9 \\
35 & -1364.075 \\
40 & -2454.8 \\
45 & -3714.075 \\
50 & -5141.9 \\
\hline
\end{array}
\][/tex]
### Sketching the Graph:
1. Draw Axes:
- The x-axis represents tire pressure ([tex]\( x \)[/tex]) ranging from 0 to 50.
- The y-axis represents mileage in thousands of miles.
2. Plot Function Points:
Use the table to plot points and draw the parabolic curve accurately.
3. Plot Given Data Point:
Plot the point [tex]\( (28, 43) \)[/tex] on the same graph. This point should be marked distinctly, such as with a different color or shape.
4. Label the Graph:
Add labels to the axes, give the graph a title, and optionally include a legend to differentiate the function line and the data points.
### Conclusion:
The visual representation of the graph lets you see how the actual data point (28, 43) compares with the values calculated using the model function. The sketch should help you understand the relationship between tire pressure and mileage and how well the data point fits the modeled function.
If you need actual plots for checking correctness, you could use graphing utilities, but this detailed process is how to approach it step-by-step.
### Step-by-Step Solution:
1. Understand the Function:
The function given is [tex]\( f(x) = -0.589 x^2 + 38.1 x - 561 \)[/tex]. This formulates a parabola. Since the coefficient of [tex]\( x^2 \)[/tex] is negative, the parabola opens downwards.
2. Mileage Data Provided:
The data provided in the table is for [tex]\( x = 28 \)[/tex] pounds per square inch, and the mileage is 43 thousand miles.
3. Plotting the Function:
To sketch the graph, we need to plot the function over a range of [tex]\( x \)[/tex]-values. Typically, [tex]\( x \)[/tex]-values of tire pressure can range from 0 to 50 pounds per square inch.
4. Calculate [tex]\( f(x) \)[/tex] Values for a Range of [tex]\( x \)[/tex]:
We need to evaluate the function [tex]\( f(x) \)[/tex] at various [tex]\( x \)[/tex]-values within our chosen range to get the shape of the parabola.
5. Plot the Function and the Data Point:
- For each [tex]\( x \)[/tex]-value in our range, compute [tex]\( f(x) \)[/tex].
- Plot these points to visualize the function.
- Plot the specific data point given: [tex]\( (28, 43) \)[/tex].
### Creating a Table of Values:
Let's create a table of values for [tex]\( x \)[/tex] from 0 to 50 with an interval of 5, and calculate the corresponding [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & -561 \\
5 & 132.825 \\
10 & 557 \\
15 & 711.525 \\
20 & 596.4 \\
25 & 211.625 \\
30 & -441.9 \\
35 & -1364.075 \\
40 & -2454.8 \\
45 & -3714.075 \\
50 & -5141.9 \\
\hline
\end{array}
\][/tex]
### Sketching the Graph:
1. Draw Axes:
- The x-axis represents tire pressure ([tex]\( x \)[/tex]) ranging from 0 to 50.
- The y-axis represents mileage in thousands of miles.
2. Plot Function Points:
Use the table to plot points and draw the parabolic curve accurately.
3. Plot Given Data Point:
Plot the point [tex]\( (28, 43) \)[/tex] on the same graph. This point should be marked distinctly, such as with a different color or shape.
4. Label the Graph:
Add labels to the axes, give the graph a title, and optionally include a legend to differentiate the function line and the data points.
### Conclusion:
The visual representation of the graph lets you see how the actual data point (28, 43) compares with the values calculated using the model function. The sketch should help you understand the relationship between tire pressure and mileage and how well the data point fits the modeled function.
If you need actual plots for checking correctness, you could use graphing utilities, but this detailed process is how to approach it step-by-step.
