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?

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]

3.2: 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 |

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

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, ∞)

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

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?

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.

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]

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

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

3.2: Graphs of Functions Assignment Page 161 19-27, 33-39 (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”

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?

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

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}

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)

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 – 6 + 4 = 2x - 2 , x > 3 -2x + 6 + 4 = -2x +10 , x < 3 2x – 6 = 0 +6 +6 2x = 6 x = 3

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

3.2: Graphs of Functions Assignment Page 161 41 - 53 (odd problems)

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?

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

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)

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

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