1. Chapter 3: Functions and Graphs 3.2: Graphs of Functions
Essential Question: What can you look for in a graph to determine if the graph represents a function?

2. 3.2: Graphs of Functions Ex 1: Functions Defined by Graphs
A graph may be used to define a function or relation. Suppose that the graph below defines a function f.
Find:
f (0)
f (3)
f (2)
The domain of f
The range of f
f (0) = 7
f (3) = 0
f (2) = undefined
[-8, 2) and (2, 7]
[-9, 8]

3. 3.2: Graphs of Functions Ex 2: The Vertical Line Test
A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.
Not a Function Function |

4. 3.2: Graphs of Functions
Ex 3: Where a Function is Increasing/Decreasing
A function is said to be increasing on an interval if its graph always rises as you move left to right.
It is decreasing if its graph always falls as you move left to right
A function is said to be constant on an interval if its graph is a horizontal line over the interval

5. 3.2: Graphs of Functions
Ex 3: Where a Function is Increasing/Decreasing
On what interval is the function f (x) = |x| + |x – 2| increasing? Decreasing? Constant?
Graph the function
It suggests that f is
(For now) Only use brackets if an interval is constant
Decreasing from (-∞, 0)
Constant on [0, 2]
Increasing on (2, ∞)

6. 3.2: Graphs of Functions Assignment Page 160
1 – 14, 17 & 18 (all problems)

7. Chapter 3: Functions and Graphs 3.2: Graphs of Functions Day 2
Essential Question: What can you look for in a graph to determine if the graph represents a function?

8. 3.2: Graphs of Functions Ex 4: Finding Local Maxima and Minima
A graph of a function may include some peaks and valleys.
The peak may not be the highest point, but it is the highest point in its area (called a local maximum)
A valley may not be the lowest point, but it is the lowest point in its area (called a local minimum)
Calculus is usually needed to find exact local maxima and minima. However, they can be approximated with a calculator.

9. 3.2: Graphs of Functions Ex 4: Finding Local Maxima and Minima
Graph f (x) = x3 – 3.8x2 + x + 1 and find all local maxima and minima.
Graph is shown on calculator
You can find local maxima and minima by using the FMIN and FMAX just like finding the root from a graph.
[Graph] → [more] → [math] → [fmin]/[fmax]

10. 3.2: Graphs of Functions Ex 5: Analyzing a Graph
Concavity and Inflection Points
A point where the curve changes concavity is called an inflection point
An inflection point will be always be between a local maximum and local minimum’s x-values
Concavity is used to describe the way a curve bends
Connect two points on a curve, between inflection points
If the line is above the curve, it’s concave up
If the line is below the curve, it’s concave down
Open up = concave up, open down = concave down

11. 3.2: Graphs of Functions Ex 5: Analyzing a Graph
Graph the function f (x) = -2x3 + 6x2 – x + 3
Find
All local maxima and minima of the function
Intervals where the function is increasing/decreasing
All inflection points of the function
Intervals where the function is concave up and where it is concave down

12. 3.2: Graphs of Functions Assignment Page 161
19-27, (odd problems)
Hint #1: Do problems 23 – 27 before 19 & 21
Hint #2: For 33 – 35, find the inflection point first
Hint #3: For 37 & 39:
I don’t need to see your graph (part “a”)
Find part “c” before part “b”
Find part “e” before part “d”

13. Chapter 3: Functions and Graphs 3.2: Graphs of Functions Day 3
Essential Question: What can you look for in a graph to determine if the graph represents a function?

14. 3.2 Graphs of Functions Ex 6: Graphing a Piecewise Function
To graph a piecewise function by hand
Sketch (lightly) each of the graphs
Use the individual domain rule to only use the specified part of the graph & put them together
To graph a piecewise function on the calculator
Enter the function in normally
Divide it by the domain of its piece
Inequality symbols are in the test menu (2nd, 2)
Compound inequalities must be split up

15. 3.2: Graphs of Functions
Ex 6: Graphing a Piecewise Function (calculator)
Graph
On the graphing calculator (TI-86):
x2/(x<1)
x+2/((1<x)(x<4))
On iPhone apps (Desmos)
{x<1 : x2, 1<x<4: x+2}

16. 3.2: Graphs of Functions Ex 7: The Absolute-Value Function
Graph f (x)=|x|
This is also a piecewise function
For the first equation, flip the sign on all terms that were inside the absolute value signs.
Domain is split where the stuff inside the absolute value would equal 0 (the x-coordinate of the vertex of the absolute value function)

17. 3.2: Graphs of Functions Ex 7: The Absolute-Value Function #2
Graph f (x)=|2x – 6| + 4
What are the two equations?
Where do the equations split? (Where’s the vertex?)
2x – = 2x - 2
, x > 3
-2x = -2x +10
, x < 3
2x – 6 = 0
2x = 6
x = 3

18. 3.2: Graphs of Functions Ex 8: The Greatest Integer Function
Graph f (x)=[x]
We enter the function in as “int x”
Doesn’t look quite right, does it?
To change graphing type
(Only necessary for the greatest integer function)
On the screen to enter functions, press more
Press F3 for “Style”, use the (dot display) setting

19. 3.2: Graphs of Functions
Assignment
Page 161
(odd problems)

20. Chapter 3: Functions and Graphs 3
Chapter 3: Functions and Graphs 3.2: Graphs of Functions Uncovered This Year
Essential Question: What can you look for in a graph to determine if the graph represents a function?

21. 3.2: Graphs of Functions Ex 9: Parametric Graphing
In parametric graphing, both the x and y coordinate are given functions to a 3rd variable, t.
Graph the curve given by
x=2t + 1
y = t2 – 3
Solution, make a table of values, and sketch

22. 3.2: Graphs of Functions Ex 9: Parametric Graphing x=2t + 1 y = t2 – 3
Now graph t
x = 2t + 1
y = t2 - 3
(x, y) -2 -3 1
(-3, 1) -1
(-1, -2)
(1, -3) 3
(3, -2) 2 5
(5, 1) 7 6
(7, 6)

23. 3.2: Graphs of Functions Ex 10: Graphing (w/ calc) in parametric mode
Change mode (2nd, mode) to “Param” (5th down)
Now when you go to graph, y(x) is changed to E(t)
You also now enter in two functions at a time (x & y)
To graph y = f (x) in parametric mode
Let x = t and y = f (t)
To graph x = f(y) in parametric mode
Let y = t and x = f (t)
Alter your window
Change the t-step = 0.1

24. 3.2: Graphs of Function Ex 10: Graphing in Parametric Mode Graph
Let x = t and
Graph x = y2 – 3y + 1
Let y = t and x = t2 – 3t + 1