Question

You can refresh your memory about regression lines and the correlation coefficient by doing the MyApplet Exercises at the end of Chapter \(3 .\) a. Graph the line corresponding to the equation \(y=0.5 x+3\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line. b. Check your graph using the How a Line Works applet.

Short answer

2. What is the y-intercept of the line? 3. What is the slope of the line?

Step-by-step solution

Graph the line

To graph the line, we can use the given equation, \(y=0.5x+3\), to find the y-value corresponding to each x-value. Then, we can plot these points on the coordinate plane: - For \(x=0\), we have \(y=0.5(0)+3=3\) - For \(x=1\), we have \(y=0.5(1)+3=3.5\) - For \(x=2\), we have \(y=0.5(2)+3=4\) Plot these points on the coordinate plane and draw a straight line passing through the points.

Identify the y-intercept and slope

The y-intercept is the point where the line intersects the y-axis, and this occurs when \(x=0\). Using our results from Step 1, the y-intercept is at the point \((0,3)\). The slope of the line can be found using the equation \(y=0.5x+3\). Since the line is in slope-intercept form (i.e., \(y=mx+b\)), the slope is the coefficient of the x term, which in this case is \(0.5\).

Part a Summary

The line corresponding to the equation \(y=0.5x+3\) passes through the points \((0,3)\), \((1,3.5)\), and \((2,4)\). The y-intercept is at the point \((0,3)\), and the slope is \(0.5\).

Check your work using an applet

Open the How a Line Works applet and enter the equation of the line, \(y=0.5x+3\). The applet should show a visualization of the line, its slope, and its y-intercept. Confirm that the points \((0,3)\), \((1,3.5)\), and \((2,4)\) are on the line, and that the y-intercept and slope match the values found in Steps 1 and 2.

Key concepts

Graphing

Graphing is a fundamental concept when dealing with linear regression, as it allows us to visualize mathematical relationships. When you graph a line, you're essentially plotting points on a coordinate plane. For a specific linear equation like \( y = 0.5x + 3 \), the task begins with determining points that lie on this line.
To do this, you can select a set of \( x \)-values, substitute them into the equation, and solve for the corresponding \( y \).
For example:

• When \( x = 0 \), \( y = 0.5(0) + 3 = 3 \).
• When \( x = 1 \), \( y = 0.5(1) + 3 = 3.5 \).
• When \( x = 2 \), \( y = 0.5(2) + 3 = 4 \).

Once these points \((0,3)\), \((1,3.5)\), and \((2,4)\) are plotted, you then connect them with a straight line. This line represents all possible solutions to the equation, forming a visual representation of the relationship between \( x \) and \( y \).
Graphing helps in understanding the nature of the equation and predicting values.

Y-Intercept

In linear equations, the y-intercept represents the point where the line crosses the y-axis. It is crucial because it provides the starting value of \( y \) when \( x \) is zero. With the equation \( y = 0.5x + 3 \), when you set \( x = 0 \), you find that \( y = 3 \).
This means the y-intercept is the point \((0,3)\).
Knowing the y-intercept helps in graphing the equation accurately, as it gives a fixed point on the graph. The intercept will remain the same irrespective of changes to the slope. Therefore, it's a reliable reference point whenever you're plotting or analyzing the graph of a line.

Slope

The concept of slope is essential in understanding how a line behaves on a graph. The slope tells us how much \( y \) changes for a unit change in \( x \). In the equation \( y = 0.5x + 3 \), the slope is represented by the coefficient of \( x \), which is \( 0.5 \).
This implies that for every one unit increase in \( x \), the value of \( y \) increases by 0.5 units.
A positive slope, like 0.5, indicates that the line is rising as you move from left to right. Conversely, a negative slope would mean the line is falling. Understanding the slope is vital for predicting trends and relationships in data sets. It shows how one variable affects another and can provide insights into cause and effect within your graphed data.