1.5 GEOMETRIC PROPERTIES OF LINEAR FUNCTIONS

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Marjorie McCormick
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1 Functions Modeling Change: 1.5 GEOMETRIC PROPERTIES OF LINEAR FUNCTIONS

2 Interpreting the Parameters of a Example 1 Linear Function With time, t, in years, the populations of four towns, P A, P B, P C, and P D, are given by the following formulas: P A = 20, t, P B = 50, t, P C = 650t + 45,000, P D = 15,000(1.07) t. (a) Which populations are represented by linear functions? Functions Modeling Change:

3 Solution (a)the populations of towns A, B, and C are represented by linear functions because they can be written in the form P = b + m t. Town D s population does not grow linearly since its formula P D = 15,000(1.07) t, cannot be expressed in the form P D = b + m t. Functions Modeling Change:

4 Functions Modeling Change: Example 1 Interpreting the Parameters of a continued Linear Function P A = 20, t, P B = 50, t, P C = 650t + 45,000. (b) Describe in words what each linear model (towns A, B and C) tells you about that town s population. Which town starts out with the most people? Which town is growing fastest?

5 Functions Modeling Change: Solution Interpreting the Parameters of a Linear Function For town A: P A = 20, t b = 20,000 and m = that in year t = 0, town A initially has 20,000 people growing by 1600 people per year.

6 Functions Modeling Change: Solution Interpreting the Parameters of a Linear Function For town B: P B = 50, t b = 50,000 and m = 300. This means that town B starts with 50,000 people. The negative slope indicates the population is decreasing at the rate of 300 people per year.

7 Functions Modeling Change: Solution Interpreting the Parameters of a Linear Function For town C: P C = 650t + 45,000 b = 45,000 and m = 650. This means that town C begins with 45,000 people and grows by 650 people per year. Town B starts with the most. Town A grows the fastest.

8 Functions Modeling Change: The Effect of the Parameters on the Graph of a Linear Function Let y = b + m x. Then the graph of y against x is a line. The y-intercept, b, tells us where the line crosses the y- axis. If the slope, m, is positive, the line climbs from left to right. If the slope, m, is negative, the line falls from left to right. The slope, m, tells us how fast the line is climbing or falling. The larger the magnitude of m (either positive or negative), the steeper the graph of f.

9 Example 3 Intersection of Lines The cost in dollars of renting a car for a day from three different rental agencies and driving it d miles is given by the following functions: C 1 = d C 2 = d C 3 = 0.50d. (a)describe in words (interpret) the daily rental arrangements made by each of these three agencies. Functions Modeling Change:

10 Functions Modeling Change: Intersection of Lines C 1 = d C 2 = d C 3 = 0.50d. Solution: Agency 1 charges $50 plus $0.10 per mile. Agency 2 charges $30 plus $0.20 per mile. Agency 3 charges $0.50 per mile.

11 Functions Modeling Change: Intersection of Lines Example 3 C 1 = d C 2 = d C 3 = 0.50d. (b) Which agency is cheapest? Solution: To determine which agency is cheapest, graph all three.

13 First Window

14 Second Window

15 Third window

16 Algebraically calculate the intersection points

17 Example 3 - continued Intersection of Lines C 1 = d C 2 = d C 3 = 0.50d. C $ (125, 62.50) (200,70) (100,50) Based on the points of intersection and minimizing C, Agency 3 is cheapest for d < 100 Agency 2 is cheapest for 100 < d < 200 Agency 1 is cheapest for d > 200 d = 200? d miles What happens at d = 100 and Functions Modeling Change:

y = k is a horizontal line and its slope is zero.

18 Functions Modeling Change: Equations of Horizontal and Vertical Lines For any constant k: The graph of the equation y = k is a horizontal line and its slope is zero. The graph of the equation x = k is a vertical line and its slope is undefined.

19 Functions Modeling Change: Slopes of Parallel and Perpendicular Lines Let L 1 and L 2 be two lines having slopes m 1 and m 2, respectively. Then: These lines are parallel if and only if m 1 = m 2. These lines are perpendicular if and only if m 1 = 1/m 2.

20 Functions Modeling Change: 1.6 FITTING LINEAR FUNCTIONS TO DATA

21 v, viscocity (lbsˑsec/in2 Functions Modeling Change: Laboratory Data: The Viscosity of Motor Oil Table and graph showing relationship between viscosity and temperature T, temperature ( F) v, viscosity (lbˑsec/in 2 ) Viscosity and Temperature T, temperature (F ) The scatter plot of the data in the above figure shows that the viscosity of motor oil decreases, approximately linearly, as its temperature rises.

22 v, viscocity (lbˑsec/in2 Functions Modeling Change: The Viscosity of Motor Oil: Regression Line Using a computer or calculator to find the line of best fit Viscosity as a Function of Temperature Regression line v = T T, temperature ( F) Notice that none of the data points lie exactly on the regression line, although it fits the data well.

23 Generate the Regression Line and the Regression Equation Enter the Data into Lists

24 VARS Y-VARS FUNCTION Yx

25 Example 1 Interpolation Using the regression line v = T, predict the viscosity of motor oil at 240F⁰. Solution: At T = 240F⁰, the formula for the regression line predicts that the viscosity of motor oil is v = (240) = 5.3 lb sec/in 2. This is reasonable. The figure on the previous slide shows that the predicted point is consistent with the trend in the data points from the Table Functions Modeling Change:

26 Example 1 continued Extrapolation Using the regression line v = T, predict the viscosity of motor oil at 300 F. Solution: At T = 300 F the regression-line formula gives v = = 12.3 lb sec/in 2. This is unreasonable because viscosity cannot be negative. To understand what went wrong, notice that the point (300, 12.3) is far from the plotted data points. By making a prediction at 300F⁰, we have assumed incorrectly that the trend observed in laboratory data extended as far as 300 F⁰. Functions Modeling Change:

27 Evaluate and Interpret Y1(205) Y1(350)

28 Functions Modeling Change: Correlation The calculator also generated a correlation coefficient, r. This number lies between 1 and +1 and measures how well a particular regression line fits the data.

30 Positive Linear Correlation

31 Negative correlation

32 No correlation

33 Other correlations We Investigate LINEAR Only for now

34 How strong is the correlation?

35 Interpreting the Values of r Perfect Positive Correlation If r = 1 the data lie exactly on a line of positive slope

36 Interpreting the Values of r Perfect Positive Correlation If r = -1 the data lie exactly on a line of negative slope

37 Interpreting the Values of r r = 0 no correlation

38 Possible Values of r 1 r 1 strong correlation weak correlation r = -1 r = 0 r = +1 l r = r = r = 0.07 r = 0.69

39 Functions Modeling Change: Correlation For Example 1, r is negative because the slope of the line is negative. r being close to -1 indicates that the regression line is a good fit, therefore we have high confidence in interpolation predictions.

40 The Difference between Relation, Correlation, and Causation It is important to understand that a high correlation (either positive or negative) between two quantities does not imply causation. For example, there is a high correlation between children s reading level and shoe size. However, large feet do not cause a child to read better (or vice versa). Larger feet and improved reading ability are both a consequence of growing older. A correlation of r = 0 usually implies there is no linear relationship between x and y, but this does not mean there is no relationship at all. Functions Modeling Change:

Section Functions and Function Notation

Section Functions and Function Notation Section 1.1 - Functions and Function Notation A function is a relationship between two quantities. If the value of the first quantity determines exactly one value of the second quantity, we say the second