Problem 32 - College Algebra with Corequisite Support

Question

Graph \(f(x)=0.5 x+10 .\) Pick a set of five ordered pairs using inputs \(x=-2,1,5,6,9\) and use linear regression to verify that the function is a good fit for the data.

Step-by-Step Solution

Answer

The function \(f(x) = 0.5x + 10\) is a perfect fit for the data points.

Step 1: Calculate Output Values

First, calculate the output values for the given input values using the function \(f(x) = 0.5x + 10\).For \(x = -2\):\[f(-2) = 0.5(-2) + 10 = 9\]For \(x = 1\):\[f(1) = 0.5(1) + 10 = 10.5\]For \(x = 5\):\[f(5) = 0.5(5) + 10 = 12.5\]For \(x = 6\):\[f(6) = 0.5(6) + 10 = 13\]For \(x = 9\):\[f(9) = 0.5(9) + 10 = 14.5\]The ordered pairs are: \((-2, 9), (1, 10.5), (5, 12.5), (6, 13), (9, 14.5)\).

Step 2: Plot the Ordered Pairs

Plot the ordered pairs \((-2, 9), (1, 10.5), (5, 12.5), (6, 13), (9, 14.5)\) on a graph. This will provide a visual representation of how the data points align with the linear function \(f(x) = 0.5x + 10\).

Step 3: Perform Linear Regression

Apply linear regression to check if the function \(f(x) = 0.5x + 10\) is a good fit. Linear regression involves calculating the line of best fit's parameters (like slope and intercept) based on the given data points. In this specific case, since the line is already defined by the function, verify if the calculated slope (about 0.5) and intercept (about 10) from the linear regression match the function's parameters.

Step 4: Verify Function Accuracy

Check the results from the linear regression. Since the data points align exactly with the function \(f(x) = 0.5x + 10\), the slope and intercept should perfectly match \(m = 0.5\) and \(b = 10\) respectively. Any deviations would suggest the function isn't a perfect fit, but in this case, the outcome should confirm the function is an ideal fit. This validates the function as a good representation of the data.

Key Concepts

Ordered Pairs

Understanding ordered pairs is a fundamental concept in coordinate geometry, especially when dealing with graphs of equations. Each ordered pair is a point on the Cartesian plane and consists of an "x-coordinate" and a "y-coordinate". In the ordered pair \(x, y\), \(x\) represents the horizontal position, while \(y\) corresponds to the vertical position.

In the exercise, we calculated five ordered pairs using the function \(f(x) = 0.5x + 10\):

• \((-2, 9)\)
• \((1, 10.5)\)
• \((5, 12.5)\)
• \((6, 13)\)
• \((9, 14.5)\)

These ordered pairs are derived by substituting each input value into the function to get a corresponding output value. This connection between inputs and outputs is critical for graphing equations.

Graphing Linear Equations

When graphing linear equations, you represent the equation on a coordinate plane, where each point satisfies the equation. The straight line that emerges is defined by its slope and intercept.

To graph \(f(x) = 0.5x + 10\), we take the calculated ordered pairs from the previous step and plot them on a graph. These points will align to form a straight line, visually demonstrating the behavior of the linear equation.

Graphing linear equations is essential for visual learners, as it illustrates how changes in \(x\) affect \(y\). By looking at the graph, students can see the trend of data and how well the equation fits the data points. In this exercise, plotting the points revealed that they perfectly align, confirming that \(f(x) = 0.5x + 10\) is an accurate description of the relationship.

Slope and Intercept

The slope and intercept are two critical parameters that characterize a linear equation. The slope, often denoted by \(m\), measures the steepness of the line and the direction of its tilt on the graph. It is calculated as the change in \(y\) divided by the change in \(x\) between two points. A slope of 0.5, as in \(f(x) = 0.5x + 10\), indicates a gentle upward trend.

The intercept, denoted by \(b\), is the y-coordinate where the line crosses the y-axis. For the function \(f(x) = 0.5x + 10\), the intercept is 10. This means that even when \(x\) is zero, the output value starts at 10.

• The slope helps in understanding the rate of change between variables.
• The intercept shows the starting point when the independent variable is zero.

Understanding these concepts allows you to interpret the graph better and determine how well a linear equation models the data. In linear regression, verifying these parameters ensures the equation is a good fit for the given data.