Math for Liberal Arts MAT 110: Chapter 9 Notes Functions: The - - PowerPoint PPT Presentation

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Math for Liberal Arts MAT 110: Chapter 9 Notes

David J. Gisch March 10, 2012

Functions: The Building Blocks of Mathematics Linear Modeling

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. Δ Δ

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Finding the S lope of a Line

• To find the slope of a straight line, look at any two points

and divide the change in the dependent variable by the change in the independent variable.

Δ Δ

Rate of Change. S lope

Example 9.B.1: Town City had a population of 5000 in

• 1930. In 1938 the population grew to 6432. Assuming the

growth is linear, what is the rate of change?

Rate of Change. S lope

Example 9.B.2: What is the rate of change?

Slide 9-8

The Rate of Change Rule

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

tim e period. Usually a period of time.

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General Equation for a Linear Function 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 .

Equations of Lines Rate of Change. S lope

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.

Rate of Change. S lope

Example 9.B.4: DMACC hires a speaker for iWeek for $1000. It plans to sell tickets for the speaker at a rate of $5. Give a function that describes the profit/loss for DMACC.

Rate of Change. S lope

Example 9.B.5:Consider some who gets paid $12 per hour. Write a function for the amount they earn.

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Visualizing the lines

• We just created equations from a word problem. Let us

now get a better understanding of the visualization of lines.

▫ What does the rate of change/slope tell us? ▫ What does the y-intercept tell us?

S lope and Intercept

For example, the equation y = 4x – 4

• The slope is 4, and
• The y-intercept is 4

Varying the S lope

When you see slop think of how you read, left-to-right.

• Positive slopes go UP
• Negative slopes of DOWN
• A slope of zero is flat.

Varying the Intercept

The figure to the right shows the effects of changing the y- intercept for a set of lines that have the same slope. All the lines rise at the same rate, but cross the y-axis at different points.

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Rain Depth Equation

Example 9.B.6: Use the function shown in the graph to write an equation that describes the rain depth at any time after the storm began. Use the equation to find the rain depth 4 hours after the storm began. Step 1: Let x be the independent variable and y be the dependent variable. Find the change in each variable between the two given points, and use these changes to calculate the slope. Δ Δ 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.

Creating a Linear Function from Two Data Points

Linear Function

Example 9.B.7: Using the two points in the graph write the equation of the line.

Linear Function

Example 9.B.8: Suppose a dog weighed 2.5 pounds at birth and 15 pounds after one year. Give a linear function to describe the dog's growth.

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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

exponential growth (doubling) and decay (half-life). 2

⁄

1 2

⁄

• If we say the new value is a quantity , and the initial

value is an initial quantity , we can rewrite the equations as 2

⁄

1/2

⁄

• An exponential function grows (or decays) by the

same relative amount per unit time. For any quantity growing exponentially with a fractional growth rate , 1 Where *Negative values of correspond to exponential decay. *Note that the units of time used for t and r must be the same.

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Exponential Function

Example 9.C.1: China’s rapid economic development has lead to an exponentially growing demand for energy, and China generates more than two-thirds of its energy by burning coal. During the period 1998 to 2008, China’s coal consumption increased at an average rate of 8% per year, and the 2008 consumption was about 2.1 billion tons of

• coal. Use these data to predict China’s coal consumption in

2028.

Exponential Function

Example 9.C.2: When doctors prescribe medicine, they must consider how much the drugs effectiveness will decrease as time passes. If each hour a drug is only 95% as effective as the previous hour, at some point the patient will not be receiving enough medicine and must be given another dose. If the initial dose was 250 mg and the drug was administered 3 hours ago how much is left? If the drug is ineffective when below 52 mg, should the patient be given another dose?

Exponential Function

Example 9.C.3: A community of rabbits begins with an initial population of 100 and grows 7% per month. What is the population after 3 years?

Exponential Function

Example 9.C.4: The number of DMACC students doubles in every 16 years. If the population was 18,000 students in 2000, what will the population be in 2030

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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?