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Chabot Mathematics. §2.2 Graphs of Functions. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 2.1. Review §. Any QUESTIONS About §2.1 → Intro to Functions Any QUESTIONS About HomeWork §2.1 → HW-03. Recall Ordered Pairs.

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Chabot Mathematics

§2.2 Graphsof Functions

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu


Review

MTH 55

2.1

Review §

  • Any QUESTIONS About

    • §2.1 → Intro to Functions

  • Any QUESTIONS About HomeWork

    • §2.1 → HW-03


Recall ordered pairs
Recall Ordered Pairs

  • An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.

  • An ordered pair that satisfies an equation is called asolution of the equation


Ordered pair dependency
Ordered Pair Dependency

  • Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.

    • i.e., if we KNOW x, we can CALCULATE y


Mathematical relation
Mathematical RELATION

  • Any set of ordered pairs is called a relation. The set of all first (x) components is called the domain of the relation, and the set of all SECOND (y) components is called the RANGE of the relation

  • Any (x, y) Relation can be plotted on a “Cartesian” GRAPH

    • Form Fcn-Graph by Letting y = f(x)


Graph fcn f x x 2 2 x 6
Graph Fcn: f(x) = x2 – 2x – 6

  • Plot-Pts & Connect-Dots

  • Let y = f(x)


Graphing vertical line test
Graphing & Vertical-Line-Test

  • Test a Reln-Graph to see if the Relation represents a Fcn

  • If noVERTICAL lineintersects the graph of a relation at morethan one point, then the graph is the graph of a function.

FAILS Test


Example vertical line test
Example  Vertical-Line-Test

  • Use the Vertical Line Test to determine if the graph represents a function

  • SOLUTION

    • NOT a function as the Graph Does not pass the vertical line test


Example vertical line test1
Example  Vertical-Line-Test

  • Use the Vertical Line Test to determine if the graph represents a function

  • SOLUTION

    • NOT a function as the Graph Does not pass the vertical line test

TRIPLEValued


Example vertical line test2
Example  Vertical-Line-Test

  • Use the Vertical Line Test to determine if the graph represents a function

  • SOLUTION

    • IS a function as the Graph Does pass the vertical line test

SINGLEValued

SINGLEValued


Example vertical line test3
Example  Vertical-Line-Test

  • Use the Vertical Line Test to determine if the graph represents a function

  • SOLUTION

    • IS a function as the Graph Does pass the vertical line test

SINGLEValued


Example analyze fcn graph
Example  Analyze Fcn Graph

  • Let:

  • Is the point (1, −3) on the graph of f ?

  • Find all values of x such that (x, 5) is on the graph of f.

  • Find all y-intercepts of the graph of f.

  • Find all x-intercepts of the graph of f.


Example analyze fcn graph1

?

Example  Analyze Fcn Graph

  • SOLUTION: a. Is the point (1, −3) on the graph of f ?

  • (1, −3) is not on the graph of f

    • That is, (1, −3) does NOT Make this equation TRUE


Example analyze fcn graph2
Example  Analyze Fcn Graph

  • SOLUTION: b. Find all values of x such that (x, 5) is on the graph of f

    • Substitute 5 for yand solve for x.

  • (−2, 5) and (4, 5) are on the graph of f


Example analyze fcn graph3
Example  Analyze Fcn Graph

  • SOLUTION: c. Find all y-intercepts (y when x = 0) of the graph of f.

    • Substitute 0 for xand solve for y.

  • The only y-intercept is (0, −3)


Example analyze fcn graph4
Example  Analyze Fcn Graph

  • SOLUTION: d. Find all x-intercepts (x when y = 0) of the graph of f.

    • Substitute 0 for yand solve for x.

  • The x-intercepts of the graph of f are (−1, 0) and (3, 0)


Domain range from graph
Domain & Range from Graph

  • Regarding the graph of a function:

  • Domain = the set of a function’s inputs, found on the horizontal axis

    • That is, the Fcn’s X-axis Coverage

  • Range = the set of a function’s outputs, found on the vertical axis

    • That is, the Fcn’s Y-axis Coverage


Example graph domain range

Graph this function. Then estimate the domain and range.

Example  Graph Domain/Range

  • Graphing

  • Domain = [1, )

    • Covers X-axis from 1 to infinity

  • Range = [0, )

    • Covers the NON-negative portion of the Y-axis


Example statin drugs
Example  Statin Drugs

  • Many pharmaceuticals used to lower high blood cholesterol levels are called statins and are very popular and widely prescribed. These drugs, along with proper diet and exercise, help prevent heart attacks and strokes.

  • BioChemists define BioAvailability is the amount of a drug you have ingested that makes it into your bloodstream


Example statin drugs1
Example  Statin Drugs

  • A statin with a BioAvailability of 30% has been prescribed for Fernando to treat his cholesterol levels. Fernando takes 20 milligrams (20 mg) of this statin every day. During the same day, one-half of the statin is filtered OUT of the BloodStream.

  • Find the maximum concentration of the statin in the bloodstream on each of the first ten days of using the drug, and graph the result


Example statin drugs2
Example  Statin Drugs

  • SOLUTION

    • Since the statin has 30% bioavailability and Fernando takes 20 milligrams per day, the maximum concentration in the bloodstream is 30% of 20 mg, or 20(0.3) = 6 mg from each day’s prescription. Because one-half of the statin is filtered out of the blood each day, the daily maximum concentration is


Example statin drugs3

Make T-Table using Eqn:

MaxConcen = ½(Previous Max) + 6

Example  Statin Drugs


Example statin drugs4

Graph T-table

Example  Statin Drugs

  • Find Answer by Analyzing Graph

    The Maximum Statin Concentration Approaches 12 mg.


Whiteboard work
WhiteBoard Work

  • Problems From §2.2 Exercise Set

    • PPT → 50, 82

    • 18, 52

  • Alabama AutoAccident Rates by County


Prob 2 2 50
Prob 2.2-50

ID two ages for which drivers have the same number of Accidents


P2 2 50
P2.2-50

  • Draw Horizontal Line at GUESSED, or Estimated, location then find ages

22

68


P2 2 501
P2.2-50

  • Check Graph Estimates in Eqn

    • A  22 years

  • Check Graph Estimates in Eqn

    • A  68 years


P2 2 54 find x for f x 2 5
P2.2-54  Find x for f(x) = 2.5

ReCall for Graphing: use y = f(x)

  • Horizontal Line aty = 2.5%

(12, 2.5)

(94, 2.5)

94

12


P2 2 54 find x for f x 2 51
P2.2-54  Find x for f(x) = 2.5

  • What do These Results mean in terms of the variables in this situation?

  • The Functional Interpretation is that TWO values for the INdependent variable, x, produce the SAME value for the DEpendent variable y:


All done for today
All Done for Today

EngineeringDegreesin USA


Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu