Module 2  Lines 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 2.3: Slopes and Rates of Change 
The slope and yintercept of the bestfit line are helpful in understanding a set of data and the relationship that exists between the quantities in the set. This lesson explores the meaning of slope and yintercept in the context of the carvalue problem that was introduced in Lesson 2.2. SlopeIntercept Form of a Line Any nonvertical line can be written in the form where m is the slope of the line and b is the yintercept. This form of a linear equation is called the "slopeintercept" form of a line. Changing PointSlope Form to SlopeIntercept Form In Lesson 2.1 you found two pointslope equations that represent the line through the points (1, 2) and (2, 3). The equations were Each equation can be changed into slopeintercept form by performing the indicated multiplication and combining the constant terms. Notice that both equations are transformed into the same slopeintercept equation. Even though the pointslope forms appear different, the slopeintercept forms are identical. If any point on this line is used to write a pointslope form for the equation, it will simplify to the same slopeintercept form. Slope and yIntercept Values The slope and yintercept values indicate characteristics of the relationship between the two variables x and y.
Recall that the line of best fit for the car values found in Lesson 2.2 was where x represents the age of the car and y represents the car's value. The slope of the line that best fits the car data is 1478 dollars per year and the yintercept is 13,906 dollars. 2.3.1 What does the slope of the line of best fit tell you about the change in the value of this model of car? What does the yintercept tell you about the value of the car? Click here for the answer. The Units of Slope
The slope of a line is the change in y,
y (read "delta y"), divided by the change in x,
x (read "delta x"). That is, slope is
, which is a rate of change. Stating the units for
y and
x will help clarify what the slope tells you about how the yvalue changes from one xvalue to the next. In the previous example, stating the slope as
is more descriptive than just stating that the slope is 2.3.2 The linear equation that describes the value of an average home is y = 5632x + 14760, where x is years since 1970 and y is the value of an average house in dollars. What does the slope of this line tell you about the value of an average house in this area? What does the y intercept tell you? Click here for the answer. Value of a House as a Function of Size The real estate section of a local newspaper listed the selling price along with the size in square feet for houses that are for sale in the area. The scatter plot for the data looked approximately linear and the linear regression equation for the data was found to be where x is the size in square feet and y is the price of the house. 2.3.3 What does the slope of this line tell you about the homes listed in this area? What does the yintercept tell you? Click here for the answer. 100Meter Dash Carl and Ray compete in a 100meter dash. Carl's distance from the starting line after the start of the race can be modeled by y = 10.7x, where x represents time in seconds and y is measured in meters. The corresponding equation for Ray is y = 8.1x + 10. Below are the graphs for the two runners in a [0, 10, 1] x [10, 120, 10] Viewing window. 2.3.4 What do the slopes tell you about the two runners? What do the yintercepts tell you about the start of the race? Who won? Click here for the answer. The Line of Best Fit as an Approximation The equation that is created when you perform linear regression is an equation for the linear model that produces the least amount of error between the actual output values (yvalues) in the data set and the output values predicted by the model for those input values (xvalues). Because the line only approximates the data, many of the data points may not lie on the line and the model may imply erroneous conclusions. For example, the line that represents Ray's distance from the starting line indicates that he started the race 10 meters ahead of the starting line. It's quite possible that he started the race at the starting line but the model needed to have a yintercept of 10 to attain the least error between the model and the actual data over the entire data set. 
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