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Chapter 3: Functions and Graphs 3.2: Graphs of Functions

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

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Chapter 3: Functions and Graphs 3.2: Graphs of Functions

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  1. Chapter 3: Functions and Graphs3.2: 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 increasingon 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 • 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 Graphs3.2: Graphs of FunctionsDay 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, 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”

  13. Chapter 3: Functions and Graphs3.2: Graphs of FunctionsDay 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: • x2/(x<1) • x+2/((1<x)(x<4))

  16. 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)

  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?) , x > 3 2x – 6 + 4 = 2x - 2 -2x + 6 + 4 = -2x +10 , x < 3 2x – 6 = 0 +6 +6 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 • 41 - 53 (odd problems)

  20. Chapter 3: Functions and Graphs3.2: Graphs of FunctionsUncovered 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

  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

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