1. Linear, Quadratic, and Exponential Functions
Math in Our World
Section 7.7
Linear, Quadratic, and Exponential Functions

2. Learning Objectives Graph linear functions. Graph quadratic functions.
Apply quadratic functions to real-world problems.
Graph exponential functions.
Apply exponential functions to real-world problems.

3. Linear Functions
A linear function is a function of the form f(x) = ax + b, where a and b are real numbers.
The function f(x) = 0.217x can be used to model the unemployment rate for 2008, with f representing the percentage of workers who are unemployed and x representing the month of the year.
For example, f(10) = 6.57; this tells us that in October (the 10th month), the unemployment rate was 6.57%.

4. EXAMPLE 1 Graphing Linear Functions
Graph each linear function.

5. EXAMPLE 1 Graphing Linear Functions
SOLUTION
In this method, we’ll take advantage of knowing the slope, which in this case is 3, and the y intercept, which is (0, – 2). Plot the point (0, – 2), and then use a rise of 3 and a run of 1 to find a second point. Then we draw the line connecting those points.

6. EXAMPLE 1 Graphing Linear Functions
SOLUTION
In the second method, we’ll evaluate the function for three x values then plot the associated points, and again draw a line connecting the points.

7. Quadratic Functions
A function of the form f(x) = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0, is called a quadratic function. The graph of a quadratic function is called a parabola.

8. Parabolas
Note that when a is positive, it opens upward, and when a is negative, it opens downward.

9. Parabolas
The point where a parabola changes direction is called the vertex.
Every parabola has two halves that are mirror images of each other, with the divider being a vertical line through the vertex, called the axis of symmetry.
We say that a parabola is symmetric about its axis of symmetry.

10. Parabolas Procedure For Graphing a Parabola
Step 1 Identify a, b, and c; then use the formula x = – b/2a to find the x coordinate of the vertex.
The y coordinate can then be found by substituting the x value into the function.
Step 2 Find the y intercept by evaluating the function for x = 0.
Step 3 Find the x intercepts, if any, by substituting 0 for f(x) and solving for x, using either factoring or the quadratic formula.

11. Parabolas Procedure For Graphing a Parabola
Step 4 If there are no x intercepts, find at least one other point on each side of the vertex to help determine the shape.
Step 5 Plot all the points you found, then connect them with a smooth curve.

12. EXAMPLE 2 Graphing a Quadratic Function
Graph the function f(x) = x2 – 6x + 5.
SOLUTION
Step 1 In this case, a = 1, b = – 6, and c = 5. We can begin by finding the x coordinate of the vertex:
Evaluate f(3) to find the y coordinate:
The vertex is (3, – 4).

13. EXAMPLE 2 Graphing a Quadratic Function
SOLUTION
Step 2 Find the y intercept by evaluating f(0).
The y intercept is (0, 5).
Step 3 Find the x intercepts by substituting 0 for f(x) and solving.

14. EXAMPLE 2 Graphing a Quadratic Function
SOLUTION
Step 4 We already have at least one point on each side of the vertex, so this should be enough to draw the graph.
Step 5 Plot all of the points we found: the vertex (3, – 4), and intercepts (0, 5), (5, 0), and (1, 0). Then connect them with a smooth curve, making sure to change direction at the vertex.

15. EXAMPLE 3 Modeling Demographic Data with a Quadratic Function
The percentage of the population in the United States that was foreign-born can be modeled by the function P(x) = x2 – 0.420x , where x is the number of years after (a) When did the percentage reach its low point? (b) What was the percentage in 2008? (c) If the model accurately predicts future trends, what will the percentage be in 2020?

16. EXAMPLE 3 Modeling Demographic Data with a Quadratic Function
SOLUTION
(a) The graph of this function is a parabola that opens upward, since a is positive, so the lowest point occurs at the vertex.
The vertex occurs at about x = 64, or 64 years after The low point for the percentage was in 1964.
(b) The year 2008 is 108 years after 1900, and corresponds to x = 108.
In 2008, about 12.6% of the population was foreign-born.

17. EXAMPLE 3 Modeling Demographic Data with a Quadratic Function
SOLUTION
(c) The year 2020 corresponds to x = 120.
The model predicts that 16.6% of the population will be foreign-born in 2020.

18. EXAMPLE 4 Fencing a Yard Efficiently
A family buys a new puppy, and plans to fence in a rectangular portion of their backyard for an exercise area. They buy 60 feet of fencing, and plan to put the fence against the house so that fencing is needed on only three sides. Find a quadratic function that describes the area enclosed, and use it to find the dimensions that will enclose the largest area.

19. EXAMPLE 4 Fencing a Yard Efficiently
SOLUTION
In this situation, a diagram will be very helpful. If we let x = the length of the sides touching the house, then the side parallel to the house will have length 60 – 2x (60 feet total minus two sides of length x).
The area of a rectangle is length times width, so in this case, we get
A(x) = x(60 – 2x)

20. EXAMPLE 4 Fencing a Yard Efficiently
SOLUTION
To find a, b, and c, we should multiply out the parentheses:
A(x) = 60x – 2x2 or A(x) = – 2x2 + 60x
So a = – 2, b = 60 and c = 0.
The maximum area will occur at the vertex of the parabola:
The sides touching the house should be 15 feet long. The side parallel to the house should be 60 – 2(15) = 30 feet long.

21. Exponential Functions
An exponential function has the form f(x) = ax, where a is a positive real number, but not 1.
Examples of exponential functions are

22. Exponential Functions
The graph of an exponential function has two forms.
1. When a > 1, the function increases as x increases.

23. Exponential Functions
The graph of an exponential function has two forms.
2. When 0 < a < 1, the function decreases as x increases.

24. Exponential Functions
The graph of an exponential function has two forms.
For any acceptable value of a, the expression a0 = 1, so every exponential function of the form f(x) = ax has y intercept (0, 1).
Also, the graph approaches the x axis in one direction but never touches it. When this happens, we say that the x axis is a horizontal asymptote of the graph.

25. EXAMPLE 5 Graphing an Exponential Function
Draw the graph of each function.
f(x) = 2x
g(x) = 3x+1

26. EXAMPLE 5 Graphing an Exponential Function
SOLUTION
(a) Pick several numbers for x and find f(x) = 2x:
Plot the points, then connect them with a smooth curve. Note the graph approaching the x axis to the left.

27. EXAMPLE 5 Graphing an Exponential Function
SOLUTION
(a) Pick several numbers for x and find g(x) = 3x+1 :
Plot the points, then connect them with a smooth curve. Note the graph approaching the x axis to the left.

28. EXAMPLE 6 Graphing an Exponential Function
Draw the graph for
SOLUTION
(a) Pick several numbers for x and find f(x):

29. EXAMPLE 7 Exponential Population Growth
Riverside County, California, is one of the five fastest-growing counties in the United States. One estimate uses the function f(t) = A0(1.04)t to model the population, where A0 is the population at time t = 0 and t is the time in years. The population in 2007 reached 2,000,000 for the first time. Use the model to predict the population in 2027.

30. EXAMPLE 7 Exponential Population Growth
SOLUTION
Let A0 = 2,000,000 and t = 20 years (since 2027 is 20 years after our base year, 2007). Then
The population is predicted to be 4,382,246 in 2027.

31. Compound Interest
Interest on a savings account can be compounded annually, semiannually, quarterly, or daily. The formula for compound interest is given by
where A is the amount of money, which includes the principal plus the earned interest, P is the principal (amount initially invested), r is the yearly interest rate in decimal form, n is the number of times the interest is compounded a year, and t is the time in years that the principal has been invested.

32. EXAMPLE 8 Computing the Value of an Investment
Using
If $10,000 is invested at 8% per year compounded quarterly, find the value of the investment (amount) after 40 years.

33. EXAMPLE 8 Computing the Value of an Investment
SOLUTION
P = $10,000
r = 8% = 0.08
t = 40 years
n = 4, since the interest is compounded quarterly, or four times a year.
At the end of 40 years, the $10,000 will grow to $237,699.07!

34. EXAMPLE 9 Carbon Dating
Carbon-14 is a radioactive isotope found in all living things. It begins to decay when an organism dies, and one can use the proportion remaining to estimate the age of objects derived from living matter, like bones, wooden tools, or textiles. The function f(x) = A x describes the amount of carbon-14 in a sample, where A0 is the original amount and x is the number of centuries since the carbon-14 began decaying. An archaeologist claims that a recently unearthed wooden bowl is 4,000 years old. What percentage of the original carbon-14 must remain for him to make that claim?

35. EXAMPLE 9 Carbon Dating SOLUTION
We need to evaluate the given function for x = 40, since 4,000 years is 40 centuries.
The amount is times what it was when it started decaying, so there is about 61.6% of the carbon-14 left.