Understanding Graphs of Functions: Theory and Practice

Chabot Mathematics §2.2 Graphsof Functions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@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 • 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 • 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 • 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) = x2 – 2x – 6 • Plot-Pts & Connect-Dots • Let y = f(x)

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 • 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-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-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-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 • 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 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 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 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 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 • 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

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 • 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 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 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

Make T-Table using Eqn: MaxConcen = ½(Previous Max) + 6 Example  Statin Drugs

Graph T-table Example  Statin Drugs • Find Answer by Analyzing Graph The Maximum Statin Concentration Approaches 12 mg.

WhiteBoard Work • Problems From §2.2 Exercise Set • PPT → 50, 82 • 18, 52 • Alabama AutoAccident Rates by County

Prob 2.2-50 ID two ages for which drivers have the same number of Accidents

P2.2-50 • Draw Horizontal Line at GUESSED, or Estimated, location then find ages 22 68

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 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) = 2.5 • 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:

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Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

Understanding Graphs of Functions: Theory and Practice