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1.2 Analyzing Graphs of Functions In section 1.1, we learned how to determine if a set of data points or an algebraic equation was a function.
Now...
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A. Am I A Function? Ex. 2
Ex. 1
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In Section 1.1, we learned how to find the domain of a function given an algebraic equation.
Now...
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B. Determine the Domain and Range.
Ex. 1
Domain: Train your eyes to travel on the x axis.
Range:
Train your eyes to travel on the y axis.
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Ex. 2
Domain:
Range:
Ex. 3
Domain:
Range:
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C. Find the Zero's of the Function
Ex. 1
Ex. 2
4x3 ­24x2 ­x + 6
3x2 + 22x ­16
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Ex. 3 Find the zero's analytically and verify graphically.
f(x) = √(3x ­ 14) ­ 8
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D. Determine the Intervals over which a graph is increasing or decreasing.
1. A function f is increasing on any interval, if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).
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2. A function f is decreasing on any interval, if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).
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3. A function f is constant on any interval, if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) = f(x2).
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Learning Opportunity Part I:
p. 87­88 #1­29 odd, 35­49 odd (Part a only) Oct 23­12:28 PM
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E. Minima and Maxima
1. A function value f(a) is called a minimum of f if there exists an interval (x1, x2) that contains (a) such that x1 < x < x2 implies f(a) < f(x).
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2. A function value f(a) is called a maximum of f if there exists an interval (x1, x2) that contains (a) such that x1 < x < x2 implies f(a) > f(x).
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Ex. 1: Use a graphing calculator to approximate the minima and maxima of the function.
f(x) = 3x2 ­ 2x ­ 5
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F. Even and Odd Functions
1. A function y = f(x) is even if, for each x in the domain of f, f(­x) = f(x). ??What does this remind you of???
Yes...if its symmetric with the y axis!
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2. A function y = f(x) is odd if, for each x in the domain of f, f(­x) = ­f(x). ??What does this remind you of???
Yes...if its symmetric with the origin!
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Should we concern ourselves about symmetry with respect to the x axis? Why or why not?
Determine whether the graph is odd, even or neither. Verify your results graphically.
Ex. 1 f(x) = x√1­x2
Odd
Ex. 2 f(x) = x3 ­ 5
Neither
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Ex. 3 Determine the intervals over which the function is increasing, decreasing or constant and determine whether the function is even, odd or neither.
Oct 23­1:41 PM
G. Piecewise Functions
Ex. 1 Graph the function WITHOUT a graphing calculator. Determine all intervals that are increasing, decreasing or constant. Is it a function? Is it odd, even or neither? Does the function have any maxima or minima? If so, where?
f(x) = √4 + x, x < 0
√4 ­ x, x > 0
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H. Write a Linear Function
Ex. 1 Write a linear function that has the indicated function values. Represent your answer in standard form.
f(­3) = ­8 and f(1) =2
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I. Graph and determine the intervals for which
f(x) > 0.
f(x) = x2 ­ 4x
Let's check our graph! Oct 23­1:49 PM
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Learning Opportunity Part II:
p. 88­90 #51­57 odd, 67­73 odd, 77­91 odd, 95­103 odd, 105­112 all
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