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  Decreasing from (- ∞, 0) Increasing on (2, ∞) Constant on [0, 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) = x 3 – 3.8x 2 + 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) = -2x 3 + 6x 2 – 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, (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 (2 nd, 2)  Compound inequalities must be split up

3.2: Graphs of Functions Ex 6: Graphing a Piecewise Function (calculator) ▫Graph ▫On the graphing calculator:  x 2 /(x<1)  x+2/((1<x)(x<4))

3.2: Graphs of Functions Ex 7: The Absolute-Value Function ▫Graph f (x)=|x| ▫This is also a piecewise function ▫  For the second 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 – = 2x x = -2x +10 2x – 6 = x = 6 x = 3, x > 3, 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  (odd problems)

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 3 rd variable, t. ▫Graph the curve given by  x=2t + 1  y = t 2 – 3 ▫Solution, make a table of values, and sketch

3.2: Graphs of Functions Ex 9: Parametric Graphing ▫x=2t + 1 ▫y = t 2 – 3 ▫Now graph tx = 2t + 1y = t 2 - 3(x, y) -2-31(-3, 1) -2(-1, -2) 01-3(1, -3) 13-2(3, -2) 251(5, 1) 376(7, 6)

3.2: Graphs of Functions Ex 10: Graphing (w/ calc) in parametric mode ▫Change mode (2 nd, mode) to “Param” (5 th 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 1)Graph ▫Let x = t and 2)Graph x = y 2 – 3y + 1 ▫Let y = t and x = t 2 – 3t + 1