3/10/2012 Math for Liberal Arts MAT 110: Chapter 9 Notes Functions: The Building Blocks of Mathematics David J. Gisch March 10, 2012 Linear Functions A linear function has a constant rate of change and a straight-line graph. • The rate of change is equal to the slope of the graph. • The greater the rate of change, the steeper the slope. • Calculate the rate of change by finding the slope between any two points on the graph. Linear Modeling ���� �� ������ � ����� � ������ �� ��������� �������� ��� ������ �� ����������� �������� ��� � Δ� Δ� 1
3/10/2012 Finding the S lope of a Line Rate of Change. S lope • To find the slope of a straight line, look at any two points Example 9.B.1: Town City had a population of 5000 in and divide the change in the dependent variable by the 1930. In 1938 the population grew to 6432. Assuming the change in the independent variable. growth is linear, what is the rate of change? ���� �� ������ � ����� � ������ �� ��������� �������� ��� ������ �� ����������� �������� ��� � Δ� Δ� Rate of Change. S lope The Rate of Change Rule Example 9.B.2: What is the rate of change? The rate of change rule allows us to calculate the change in the dependent variable from the change in the independent variable. ��������� �������� � ���� �� ������ �� ������ �� ������ ����������� �������� Usually seen as Am ount per Usually a period of time. tim e period . Slide 9-8 2
3/10/2012 Equations of Lines Rate of Change. S lope General Equation for a Linear Function Example 9.B.3: Write a function for the following situation. You rent a boat to go out on a lake. The rental is ��������� �������� $20 plus $10 per hour. � ������� ����� � ����� �� ������������������� ��������� Algebraic Equation of a Line In algebra, x is commonly used for the independent variable and y for the dependent variable. For a straight line, the slope is usually denoted by m and the initial value, or y - intercept, is denoted by b . With these symbols, the equation for a linear function becomes � � �� � � . Rate of Change. S lope Rate of Change. S lope Example 9.B.4: DMACC hires a speaker for iWeek for Example 9.B.5:Consider some who gets paid $12 per hour. $1000. It plans to sell tickets for the speaker at a rate of Write a function for the amount they earn. $5. Give a function that describes the profit/loss for DMACC. 3
3/10/2012 Visualizing the lines S lope and Intercept • We just created equations from a word problem. Let us For example, the equation now get a better understanding of the visualization of y = 4 x – 4 lines. The slope is 4, and • ▫ What does the rate of change/slope tell us? The y -intercept is �4 • ▫ What does the y-intercept tell us? Varying the S lope Varying the Intercept When you see slop think of how The figure to the right shows you read, left-to-right. the effects of changing the y - Positive slopes go UP • intercept for a set of lines that Negative slopes of DOWN • have the same slope. All the A slope of zero is flat. • lines rise at the same rate, but cross the y -axis at different points. 4
3/10/2012 Rain Depth Equation Creating a Linear Function from Two Data Points Step 1: Let x be the independent variable and y be the Example 9.B.6: Use the function shown in the graph to dependent variable. Find the change in each variable write an equation that describes the rain depth at any time between the two given points, and use these changes to after the storm began. Use the equation to find the rain calculate the slope. depth 4 hours after the storm began. ����� � � � � � � � � � � � � � � Step 2: Substitute the slope, m , and the num erical values of x and y from either point into the equation y = m x + b and solve for the y -intercept, b . Step 3: Use the slope and the y -intercept to write the equation in the form y = m x + b . Linear Function Linear Function Example 9.B.7: Using the two points in the graph write the Example 9.B.8: Suppose a dog weighed 2.5 pounds at birth equation of the line. and 15 pounds after one year. Give a linear function to describe the dog's growth. 5
3/10/2012 Linear Function Example 9.B.9: The population of a city in 1981 was 58,000 people. In 2010 it was 78,000 people. Assuming the growth was linear, give a function to describe the growth. Exponential Modeling Exponential Doubling and Half-Life • In chapter 8 we learned the following functions for • An exponential function grows (or decays) by the exponential growth (doubling) and decay (half-life). same relative amount per unit time. For any quantity � ��� ����� � �������� �������2� � � ������ ⁄ growing exponentially with a fractional growth rate � , ��� ����� � �������� �������1 2 � � � � ���� ⁄ � � � � �1 � �� � Where • If we say the new value is a quantity � , and the initial � � ����� �� ���� � value is an initial quantity � � , we can rewrite the � � � ������� ����� �� ��� �������� ��� � � 0� equations as � � ���������� ������ ���� ��� ������� �� � �������� ⁄ � � � � �2� � � ������ � � ���� ⁄ � � � � �1/2� � � ���� *Negative values of � correspond to exponential decay. *Note that the units of time used for t and r must be the same. 6
3/10/2012 Exponential Function Exponential Function Example 9.C.1: China’s rapid economic development has Example 9.C.2: When doctors prescribe medicine, they lead to an exponentially growing demand for energy, and must consider how much the drugs effectiveness will China generates more than two-thirds of its energy by decrease as time passes. If each hour a drug is only 95% as burning coal. During the period 1998 to 2008, China’s coal effective as the previous hour, at some point the patient consumption increased at an average rate of 8% per year, will not be receiving enough medicine and must be given and the 2008 consumption was about 2.1 billion tons of another dose. If the initial dose was 250 mg and the drug coal. Use these data to predict China’s coal consumption in was administered 3 hours ago how much is left? If the drug 2028. is ineffective when below 52 mg, should the patient be given another dose? Exponential Function Exponential Function Example 9.C.3: A community of rabbits begins with an Example 9.C.4: The number of DMACC students doubles initial population of 100 and grows 7% per month. What is in every 16 years. If the population was 18,000 students in the population after 3 years? 2000, what will the population be in 2030 7
3/10/2012 Exponential Function Example 9.C.5: You buy a car for $15,500. It depreciates 12% per year. How much is the car worth after 6 years? 8
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