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

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. 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.

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…

The Identity (Linear) Function

Squaring (Quadratic) Functions

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.

The Square Root and Cube Root Functions

Absolute Value Function Definition of Absolute Value |x|

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.

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.

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.

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.

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

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

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

1. Give the equation 2. Name the Domain and Range B. C. A. D. E.

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.

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.

Combining Transformations of Graphs

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

Applying a Shift to an Equation Model The number of monthly active Facebook users F in millions from 2009 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 = 2011. Interpret your result. Use your equation in part (b) to determine the year when Facebook reached 1 billion monthly active users.

Applying a Shift to an Equation Model F(x) = 233x + 200, where x is the number of years after 2009. Evaluate F(2), and interpret your result. F(2) = 233(2) + 200 = 666 and the value x = 2 corresponds to 2009 + 2 = 2011, Facebook had 666 million monthly active users in 2011.

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 + 200 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

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

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: