171S1.3p Linear Functions, Slope, Applications
August 23, 2012
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MAT 171 Dr. Claude Moore, CFCC
Section 1.3 Linear Functions, Slope, and Applications
GeoGebra Programs for Mathematics Courses http://cfcc.edu/mathlab/geogebra/
Linear Function: Graph Two Points ­ This shows the linear functon with Graph, Slope, Y­intercept, and Slope­intercept form Equation created from two points. Click on button on slider to highlight it and use left or right arrow to change value. This changes the values of (x1, y1) and (x2, y2) to construct graph and equation. Hold down the Shift­key and use the mouse to move the grid for a better view if necessary. Click the recycle icon to reset the graph and values.
Calculate the Slope, Y­intercept, and write the Equation before viewing answer.
Section 1.3 Slope between Two Points
Section 1.3 Equations of Lines: Slope–Intercept
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Applications of Slope
Linear Functions
A function f is a linear function if it can be written as f (x) = mx + b, where m and b are constants.
The grade of a road is a number expressed as a percent that tells how steep a road is on a hill or mountain. A 4% grade means the road rises 4 ft for every horizontal distance of 100 ft.
If m = 0, the function is a constant function f (x) = b. If m = 1 and b = 0, the function is the identity function f (x) = x.
Graph line given m and b.
Copyright © 2009 Pearson Education, Inc.
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Average Rate of Change
Example
Construction laws regarding access ramps for the disabled state that every vertical rise of 1 ft requires a horizontal run of 12 ft. What is the grade, or slope, of such a ramp?
Slope can also be considered as an average rate of change. To find the average rate of change between any two data points on a graph, we determine the slope of the line that passes through the two points.
Example
The percent of travel bookings online has increased from 6% in 1999 to 55% in 2007. The graph below illustrates this trend. Find the average rate of change in the percent of travel bookings made online from 1999 to 2007.
slope
= (55 - 6) / (2007 - 1999)
The grade, or slope, of the ramp is 8.3%.
Copyright © 2009 Pearson Education, Inc.
= 49 / 8 = 6.125
So, average rate of
change is 6.125% per year.
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171S1.3p Linear Functions, Slope, Applications
Section 1.3 Linear Functions, Slope, and Applications
In Exercises 1­4, the table of data contains input output values for a function. Answer the following questions for each table. 102/2.
August 23, 2012
Section 1.3 Linear Functions, Slope, and Applications
Find the slope of the line containing the given points.
“The grapher applet gives a simple 102/6.
graphing window with easy control over the viewing window.”
102/2.
a) Is the change in the inputs x the same? 110/4.
b) Is the change in the outputs y the same? c) Is the function linear?
102/8.
Since slope of AB ≠ slope of AC, these
points do NOT form a straight line.
110/4.
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Section 1.3 Linear Functions, Slope, and Applications
Find the slope of the line containing the given points.
102/16.
Use the Linear Function Graph ­ 2pts to construct graph of the lines and find the slope.
Section 1.3 Linear Functions, Slope, and Applications
Find the slope of the line containing the given points.
103/26.
103/28.
102/20.
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Section 1.3 Linear Functions, Slope, and Applications
Determine the slope, if it exists, of the graph of the given linear equation.
Use the Linear Function Graph ­ mb to construct graph of the 103/34.
lines and find the slope.
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Section 1.3 Linear Functions, Slope, and Applications
104/?.HIV Cases. HIV, human immunodeficiency virus, spreads to 10 people every minute. It is estimated that there were about 40.3 million cases of HIV worldwide in 2005. The estimated number of cases in 1985 was about 2 million. ( Source: UNAIDS) Find the average rate of change in the number of adults and children worldwide with HIV from 1985 to 2005.
103/40.
m = (40.3 ­ 2) / (2005 ­ 1985) = 38.3 / 20 = 1.915 Average rate of change = 1.915 million HIV cases per year
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171S1.3p Linear Functions, Slope, Applications
Section 1.3 Linear Functions, Slope, and Applications
104/?. Credit­ Card Debt. From 1992 to 2006, the average household credit­ card balance has risen 172%. Use the data in the graph below to find the average rate of change in the average credit­ card balance from 1992 to 2006.
August 23, 2012
Section 1.3 Linear Functions, Slope, and Applications
104/52. Find the slope and the y­ intercept of the line with the given equation: y = 4 / 7.
104/56. Find the slope and the y­ intercept of the line with the given equation: 2x ­ 3y = 12.
Use the Linear Function Graph ­ abc to construct graph of the lines and find the slope.
Solve the equation for
y = (2/3)x - 4. The slope is 2/3;
the y-intercept is (0, -4).
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Section 1.3 Linear Functions, Slope, and Applications
Section 1.3 Linear Functions, Slope, and Applications
104/62. Find the slope and the y­intercept of the line with the given equation: f(x) = 0.3 + x
104/68. Graph equation using the slope and the y­intercept: 2x + 3y = 15
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Section 1.3 Linear Functions, Slope, and Applications
Section 1.3 Linear Functions, Slope, and Applications
105/?. Pressure at Sea Depth. The function P, given by P = (1/33)d + 1, gives the pressure, in atmospheres (atm), at a depth d, in feet, under the sea. a) Graph P. NOTE: 1 atm = 14.5 psi
104/72. Stopping Distance on Glare Ice. The stopping distance (at some fixed speed) of regular tires on glare ice is a function of the air temperature F, in degrees Fahrenheit. This function is estimated by D(F) = 2F + 115, where is the stopping distance, in feet, when the air temperature is F, in degrees Fahrenheit. a) Graph D. b) Find D(0), D(­20), D(10), and D(32). c) Explain why the domain should be restricted to [­57.5, 32].
b) Find P(0), P(5), P(10), P(33), and P(100).
P(0) = 1.0 atm = 14.5 psi
P(5) = 1.2 atm = 16.7 psi
P(10) = 1.3 atm = 18.9 psi
P(33) = 2.0 atm = 29 psi
P(100) = 4.0 atm = 58 psi
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