1. Introduction
Real-world contexts that have two variables can be represented in a table or graphed on a coordinate plane. There are many characteristics of functions and their graphs that can provide a great deal of information. These characteristics can be analyzed and the real-world context can be better understood.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

2. Key Concepts
One of the first characteristics of a graph that we can observe are the intercepts, where a function crosses the x-axis and y-axis.
The y-intercept is the point at which the graph crosses the y-axis, and is written as (0, y).
The x-intercept is the point at which the graph crosses the x-axis, and is written as (x, 0).
3.3.1: Identifying Key Features of Linear and Exponential Graphs

3. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

4. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

5. Key Concepts, continued
Another characteristic of graphs that we can observe is whether the graph represents a function that is increasing or decreasing.
When determining whether intervals are increasing or decreasing, focus just on the y-values.
Begin by reading the graph from left to right and notice what happens to the graphed line. If the line goes up from left to right, then the function is increasing. If the line is going down from left to right, then the function is decreasing.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

6. Key Concepts, continued
A function is said to be constant if the graphed line is horizontal, neither rising nor falling.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

7. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

8. Key Concepts, continued
An interval is a continuous series of values. (Continuous means “having no breaks.”) A function is positive on an interval if the y-values are greater than zero for all x-values in that interval.
A function is positive when its graph is above the x-axis.
Begin by looking for the x-intercepts of the function.
Write the x-values that are greater than zero using inequality notation.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

9. Key Concepts, continued
A function is negative on an interval if the y-values are less than zero for all x-values in that interval.
The function is negative when its graph is below the x-axis.
Again, look for the x-intercepts of the function.
Write the x-values that are less than zero using inequality notation.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

10. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

11. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

12. Key Concepts, continued
Graphs may contain extrema, or minimum or maximum points.
A relative minimum is the point that is the lowest, or the y-value that is the least for a particular interval of a function.
A relative maximum is the point that is the highest, or the y-value that is the greatest for a particular interval of a function.
Linear and exponential functions will only have a relative minimum or maximum if the domain is restricted.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

13. Key Concepts, continued
The domain of a function is the set of all inputs, or
x-values of a function.
Compare the following two graphs. The graph on the left is of the function f(x) = 2x – 8. The graph on the right is of the same function, but the domain is for
x ≥ 1. The minimum of the function is –6.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

14. Key Concepts, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs

15. Key Concepts, continued
Functions that represent real-world scenarios often include domain restrictions. For example, if we were to calculate the cost to download a number of e- books, we would not expect to see negative or fractional downloads as values for x.
There are several ways to classify numbers. The following slide lists the most commonly used classifications when defining domains.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

16. Key Concepts, continued Natural numbers 1, 2, 3, ...
Whole numbers 0, 1, 2, 3, ...
Integers , –3, –2, –1, 0, 1, 2, 3, ...
Rational numbers numbers that can be written as , where a and b are integers and b ≠ 0; any number that can be written as a decimal that ends or repeats
Irrational numbers numbers that cannot be written as , where a and b are integers and b ≠ 0; any number that cannot be written as a decimal that ends or repeats
Real numbers the set of all rational and irrational numbers
3.3.1: Identifying Key Features of Linear and Exponential Graphs

17. Key Concepts, continued
An exponential function in the form f(x) = ax, where
a > 0 and a ≠ 1, has an asymptote, or a line that the graph gets closer and closer to, but never crosses or touches.
The function in the following graph has a horizontal asymptote at y = –4.
It may appear as though the graphed line touches y = –4, but it never does.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

18. Key Concepts, continued
Fairly accurate representations of functions can be sketched using the key features we have just described.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

19. Common Errors/Misconceptions
believing that exponential functions will eventually touch or intersect an asymptote
incorrectly identifying the type of function as either exponential or linear
misidentifying key features on a graph
incorrectly choosing the domain for a function
3.3.1: Identifying Key Features of Linear and Exponential Graphs

20. Guided Practice Example 1
A taxi company in Atlanta charges $2.75 per ride plus $1.50 for every mile driven. Determine the key features of this function.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

21. Guided Practice: Example 1, continued
Identify the type of function described.
We can see by the graph that the function is increasing at a constant rate.
The function is linear.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

22. Guided Practice: Example 1, continued
Identify the intercepts of the graphed function.
The graphed function crosses the y-axis at the point (0, 2.75).
The y-intercept is (0, 2.75).
The function does not cross the x-axis.
There is not an x-intercept.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

23. Guided Practice: Example 1, continued
Determine whether the graphed function is increasing or decreasing.
Reading the graph left to right, the y-values are increasing.
The function is increasing.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

24. Guided Practice: Example 1, continued
Determine where the function is positive and negative.
The y-values are positive for all x-values greater than 0.
The function is positive when x > 0.
The y-values are never negative in this scenario.
The function is never negative.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

25. Guided Practice: Example 1, continued
Determine the relative minimum and maximum of the graphed function.
The lowest y-value of the function is This is shown with the closed dot at the coordinate (0, 2.75).
The relative minimum is 2.75.
The values increase infinitely; therefore, there is no relative maximum.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

26. Guided Practice: Example 1, continued
Identify the domain of the graphed function.
The lowest x-value is 0 and it increases infinitely.
x can be any real number greater than or equal to 0.
The domain can be written as x ≥ 0.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

27. ✔ Guided Practice: Example 1, continued
Identify any asymptotes of the graphed function.
The graphed function is a linear function, not an exponential; therefore, there are no asymptotes for this function. ✔
3.3.1: Identifying Key Features of Linear and Exponential Graphs

28. Guided Practice: Example 1, continued 28
3.3.1: Identifying Key Features of Linear and Exponential Graphs

29. Guided Practice Example 2
A pendulum swings to 90% of its height on each swing and starts at a height of 80 cm. The height of the pendulum in centimeters, y, is recorded after x number of swings. Determine the key features of this function.
Number of swings (x)
Height in cm (y) 80 1 72 2
64.8 3
58.32 5
47.24 10
27.89 20
9.73 40
1.18 60
0.14
0.02
3.3.1: Identifying Key Features of Linear and Exponential Graphs

30. Guided Practice: Example 2, continued
Identify the type of function described.
The scenario described here is that of an exponential function.
We can be certain of this because the pendulum swings at 90% of its height in each swing; also, we can see from the table that the values for y do not decrease at a constant rate.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

31. Guided Practice: Example 2, continued
Identify the intercepts of the function based on the information in the table.
The function crosses the y-axis at the point (0, 80) as indicated in the table.
The y-intercept is (0, 80).
As the x-values increase, the y-values get closer and closer to 0, but do not seem to reach 0; therefore, there is not an x-intercept.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

32. Guided Practice: Example 2, continued
Determine whether the function is increasing or decreasing.
As the x-values increase, the y-values decrease.
The function is decreasing.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

33. Guided Practice: Example 2, continued
Determine where the function is positive and negative.
The y-values are positive for all x-values greater than 0.
The function is positive when x > 0.
The y-values are never negative in this scenario.
The function is never negative.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

34. Guided Practice: Example 2, continued
Determine the relative minimum and maximum of the function.
The data in the table do not change at a constant rate; therefore, the function is not linear.
Based on the information given in the problem and the values in the table, we know that this is an exponential function.
Exponential functions do not have a relative minimum because the graph continues to become infinitely smaller.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

35. Guided Practice: Example 2, continued
The height of the pendulum never goes higher than its initial height; therefore, the relative maximum of this function is (0, 80).
3.3.1: Identifying Key Features of Linear and Exponential Graphs

36. Guided Practice: Example 2, continued
Identify the domain of the function.
The lowest x-value is 0 and it increases infinitely.
x can be any real number greater than or equal to 0, but cannot be a partial swing.
The domain is all whole numbers.
3.3.1: Identifying Key Features of Linear and Exponential Graphs

37. ✔ Guided Practice: Example 2, continued
Identify any asymptotes of the function.
The points approach 0, but never actually reach 0.
The asymptote of this function is y = 0. ✔
3.3.1: Identifying Key Features of Linear and Exponential Graphs

38. Guided Practice: Example 2, continued
3.3.1: Identifying Key Features of Linear and Exponential Graphs