1. Chapter 2: Analysis of Graphs of Functions
2.1 Graphs of Basic Functions and relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs

2. 1. Which graphs are continuous. 2
1. Which graphs are continuous? 2. If the graphs are not continuous, what the point(s) of discontinuity? D. E. F.

3. 3. Where is the function increasing. 4
3. Where is the function increasing? 4. Where is the function decreasing? 5. Where is the function constant? D. E. F.

4. Basic Functions (given the name) a. Graph, b. Equation, c
Basic Functions (given the name) a. Graph, b. Equation, c. Domain, Range
Identity (Linear) Function
Squaring (Quadratic) Functions
Cubing (Cubic) Function
Square Root
Cube Root
Absolute Value Function
We will discuss the others later in the course…

5. The Identity (Linear) Function

6. Squaring (Quadratic) Functions

7. The Cubing (Cubic) Function
f(x) = x3 increases and is continuous on its entire domain, (−∞, ∞).
The point at which the graph changes from “opening downward” to “opening upward” (the point (0, 0)) is called an inflection point.

8. The Square Root and Cube Root Functions

9. Absolute Value Function
Definition of Absolute Value |x|

10. Symmetry with Respect to the y-Axis (Even Function)
If we were to “fold” the graph of f(x) = x2 along the y-axis, the two halves would coincide exactly. We refer to this property as symmetry. Symmetry with Respect to the y-Axis If a function f is defined so that f(−x) = f(x) for all x in its domain, then the graph of f is symmetric with respect to the y-axis.

11. Symmetry with Respect to the Origin (Odd Function)
If we were to “fold” the graph of f(x) = x3 along the x− and y-axes, forming a corner at the origin, the two parts would coincide. We say that the graph is symmetric with respect to the origin.
Symmetry with Respect to the Origin
If a function f is defined so that
f(−x) = −f(x)
for all x in its domain, then the graph of f is symmetric with respect to the origin.

12. Determine Symmetry Analytically
Show analytically and support graphically that
f(x) = x3 − 4x
has a graph that is symmetric with respect to the origin.
Solution: Show that f(−x) = −f(x) for any x.

13. Symmetry with Respect to the x-Axis (Not a function!)
If we “fold” the graph of x = y2 along the x-axis, the two halves of the parabola coincide. This graph exhibits symmetry with respect to the x-axis. (Note, this relation is not a function. Use the vertical line test on its graph below.)
Symmetry with Respect to the x-Axis
If replacing y with −y in an equation results in the same equation, then the graph is symmetric with respect to the x-axis.

14. Even and Odd Functions (graphically and analytically)
Example
Decide if the functions are even, odd, or neither.
a. 6x3 − 9x b. 3x2 + 5x

15. Chapter 2: Analysis of Graphs of Functions
2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs

16. Transformations! What do a, h and k do to the graph?
𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘
a means:
Flipped over the x – axis, If a < 0
Stretch (if 𝑎 >1) or
shrink (if 0< 𝑎 <1)
h means:
Left or right
k means:
Up or down
𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘
𝑓 𝑥 =𝑎 𝑥−ℎ 3 +𝑘
𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘
𝑓 𝑥 =𝑎 3 𝑥−ℎ +𝑘
𝑓 𝑥 =𝑎 𝑥−ℎ +𝑘

17. 1. Give the equation 2. Name the Domain and RangeB. C. A. D. E.

18. Combining Transformations of Graphs
Example Describe how the graph of y = −3(x − 4)2 + 5 can be obtained by transforming the graph of y = x2. Sketch its graph.

19. Combining Transformations of Graphs
Example Describe how the graph of y = −3(x − 4)2 + 5 can be obtained by transforming the graph of y = x2. Sketch its graph.

20. Combining Transformations of Graphs

21. Recognizing a Combination of Transformations
Describe the transformations from blue to red.
Give the equation of the red graph

22. Applying a Shift to an Equation Model
The number of monthly active Facebook users F in millions from to 2013 is given by
F(x) = 233x + 200,
where x is the number of years after 2009.
Evaluate F(2), and interpret your result.
Use the formula for F(x) to write an equation that gives the number of monthly active Facebook users y in millions during the actual year x.
Refer to part (b) and find y when x = Interpret your result.
Use your equation in part (b) to determine the year when Facebook reached 1 billion monthly active users.

23. Applying a Shift to an Equation Model
F(x) = 233x + 200, where x is the number of years after
Evaluate F(2), and interpret your result.
F(2) = 233(2) = 666 and the value x = 2 corresponds to = 2011, Facebook had 666 million monthly active users in 2011.

24. Applying a Shift to an Equation Model
F(x) = 233x + 200, where x is the number of years after 2009.
Use the formula for F(x) to write an equation that gives the number of monthly active Facebook users y in millions during the actual year x.
Because 2009 corresponds to 0, the graph of F(x) = 233x should be shifted 2009 units to the right.
Replace x with (x − 2009) in the formula for F(x).
y = F(x − 2009) = 233(x − 2009) + 200

25. Applying a Shift to an Equation Model (
F(x) = 233x + 200, where x is the number of years after
Refer to part (b) and find y when x = Interpret your result.
When x = 2011, y = 233(2011 − 2009) = 666, so in the year 2011 there were about 666 million monthly active Facebook users.

26. Applying a Shift to an Equation Model
Use your equation in part (b) to determine the year when Facebook reached 1 billion monthly active users.
Solve the equation: