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Functions,
Graphs, and Limits
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2. Copyright © Cengage Learning. All rights reserved.
1.2
Graphs of Equations
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3. Objectives Sketch graphs of equations.
Find the x- and y-intercepts of graphs of equations.
Find the points of intersection of two graphs.
Use mathematical models to model and solve real-life problems.

4. The Graph of an Equation

5. The Graph of an Equation
Frequently, a relationship between two quantities is expressed as an equation. For instance, degrees on the Fahrenheit scale are related to degrees on the Celsius scale by the equation
In this section, you will study some basic procedures for sketching the graphs of such equations. The graph of an equation is the set of all points that are solutions of the equation.

6. Example 1 – Sketching the Graph of an Equation
Sketch the graph of
Solution:
One way to sketch the graph of an equation is the point-plotting method. With this method, you construct a table of values that consists of several solution points of the equation, as shown in the table below.

7. Example 1 – Solution For instance, when
cont’d
For instance, when
which implies that (0, 7) is a solution point of the equation.
From the table, it follows that (0, 7), (1, 4), (2, 1), (3, –2), and (4, –5) are solution points of the equation.

8. Example 1 – Solution
cont’d
After plotting these points, you can see that they appear to lie on a line, as shown in Figure 1.13.
The graph of the equation is the line that passes through the five plotted points.
Solution Points for y = 7 – 3x
Figure 1.13

9. Example 2 – Sketching the Graph of an Equation
Sketch the graph of
Solution:
Begin by constructing a table of values, as shown below.

10. Example 2 – Solution
cont’d
Next, plot the points given in the table, as shown in Figure 1.14(a).
Figure 1.14(a)

11. Example 2 – Solution
cont’d
Finally, connect the points with a smooth curve, as shown in Figure 1.14(b).
Figure 1.14(b)

12. Graphs of Equations
The graph shown in Example 2 is a parabola. The graph of any second-degree equation of the form
has a similar shape. If a > 0, then the parabola opens upward, as shown in Figure 1.14(b), and if a < 0, then the parabola opens downward.
Figure 1.14(b)

13. Intercepts of a Graph

14. Intercepts of a Graph
Some solution points have zero as either the x-coordinate or the y-coordinate. These points are called intercepts because they are the points at which the graph intersects the x- or y-axis.
Some texts denote the x-intercept as simply the x-coordinate of the point (a, 0) rather than the point itself. Likewise, some texts denote the y-intercept as the y-coordinate of the point (0, b). Unless it is necessary to make a distinction, the term intercept will refer to either the point or the coordinate.

15. Intercepts of a Graph
A graph has no intercepts, one intercept, or several intercepts, as shown in Figure 1.17.
No x-intercept
One y-intercept
Three x-intercepts
One y-intercept
Figure 1.17

16. Intercepts of a Graph cont’d One x-intercept Two y-intercepts
No intercepts
Figure 1.17

17. Intercepts of a Graph

18. Example 3 – Finding x- and y-Intercepts
Find the x- and y-intercepts of the graph of
Solution:
To find the x-intercepts, let y be zero and solve for x.

19. Example 3 – Solution
cont’d
Because this equation has three solutions, you can conclude that the graph has three x-intercepts:
To find the y-intercepts, let x be zero and solve for y. Doing this produces
This equation has only one solution, so the graph has one y-intercept:

20. Example 3 – Solution
cont’d
See Figure 1.18.
Figure 1.18

21. Mathematical Models
Much of your study of calculus will center around the behavior of the graphs of mathematical models.
Figure 1.28 shows the graphs of six basic algebraic equations.
(a) Linear model
(b) Quadratic model
(c) Cubic model
Figure 1.28

22. Mathematical Models
cont’d
Familiarity with these graphs will help you in the creation and use of mathematical models.
(d) Square root model
(e) Absolute value model
(f) Rational model
Figure 1.28

23. Points of Intersection

24. Points of Intersection
An ordered pair that is a solution of two different equations is called a point of intersection of the graphs of the two equations.
A common business application that involves a point of intersection is break-even analysis. The marketing of a new product typically requires an initial investment.
When sufficient units have been sold so that the total revenue has offset the total cost, the sale of the product has reached the break-even point.

25. Points of Intersection
The total cost of producing x units of a product is denoted by C, and the total revenue from the sale of x units of the product is denoted by R. So, you can find the break-even point by setting the cost C equal to the revenue R and solving for x.
In other words, the break-even point corresponds to the point of intersection of the cost and revenue graphs.

26. Example 4 – Finding a Break-Even Point
A company manufactures a product at a cost of $0.65 per unit and sells the product for $1.20 per unit. The company’s initial investment to produce the product was $10,000. Will the company break even when it sells 18,000 units? How many units must the company sell to break even?
Solution: The total cost of producing x units of the product is given by
The total revenue from the sale of x units is given by

27. Example 4 – Solution
cont’d
To find the break-even point, set the cost equal to the revenue and solve for x.

28. Example 4 – Solution
cont’d
So, the company will not break even when it sells 18,000 units. The company must sell 18,182 units before it breaks even. This result is shown graphically in Figure 1.22.
Figure 1.22

29. Example 4 – Solution
cont’d
Note in Figure 1.22 that sales less than 18,182 units correspond to a loss for the company (R < C), whereas sales greater than 18,182 units correspond to a profit for the company (R > C).

30. Points of Intersection
Two types of equations that economists use to analyze a market are supply and demand equations.
A supply equation shows the relationship between the unit price p of a product and the quantity supplied x. The graph of a supply equation is called a supply curve. (See Figure 1.23.)
A typical supply curve rises because producers of a product want to sell more units when the unit price is higher
Supply Curve
Figure 1.23

31. Points of Intersection
A demand equation shows the relationship between the unit price p of a product and the quantity demanded x. The graph of a demand equation is called a demand curve. (See Figure 1.24.)
A typical demand curve tends to
show a decrease in the quantity
demanded with each increase in price.
Demand Curve
Figure 1.24

32. Points of Intersection
In an ideal situation, with no other factors present to influence the market, the production level should stabilize at the point of intersection of the graphs of the supply and demand equations. This point is called the equilibrium point.

33. Points of Intersection
The x-coordinate of the equilibrium point is called the equilibrium quantity and the p-coordinate is called the equilibrium price. (See Figure 1.25.)
Equilibrium Point
Figure 1.25

34. Points of Intersection
You can find the equilibrium point by setting the demand equation equal to the supply equation and solving for x.

35. Mathematical Models

36. Example 5 – Using a Mathematical Model
The table shows the annual average crude oil production (in millions of barrels per day) in the United States from 2009 through 2013.

37. Example 5 – Using a Mathematical Model
A mathematical model for these data is given by
where y is the annual average crude oil production (in millions of barrels per day) and t is the year, with t = 9 corresponding to 2009.

38. Example 5 – Using a Mathematical Model
cont’d
Use a graph and a table to compare the data with the model. Use the model to estimate the crude oil production in 2014.
Solution: Figure 1.27 and the table below compare the data with the model. The model appears to fit the data well.
Figure 1.27

39. Example 5 – Solution
cont’d
Using t = 14 to represent 2014, you can estimate the crude oil production in 2014 to be
million barrels per day.