Pre-Calculus Notes: Chapter 3 The Nature of Graphs

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1 Section Families of Graphs Name: Pre-Calculus Notes: Chapter 3 The Nature of Graphs Family of graphs Parent graph A group of graphs that share similar properties The most basic graph that s transformed to create the other graphs in the family Example 1 Graph f(x) = x 3 and g(x) = -x 3. Describe how the graphs of g(x) and f(x) are related. Example Use the parent graph of y x to sketch the graph of each function. y = x 1 y = (x 1) y = (x 1) + 3 1

2 Example 3 Graph each function. Then describe how it is related to the parent graph. g(x) = 4x -1 h(x) = -0.5x Section 4: Inverse Functions and Relations Inverse Relations Relations are inverses if and only if, one relation contains (a, b) whenever the other contains (b, a). Inverse Notation Original function: f(x) Inverse function: f -1 (x), read as f inverse of x Geometric Relationship Between Inverses Determine Inverses Algebraically To prove that two functions are inverses Symmetry over the line y = x Switch x and y and solve for y Show that f(g(x)) = g(f(x)) = x Example 1 Graph f(x) = x and its inverse.

3 Horizontal Line Test If every horizontal line intersects the graph of the relation in at most one point, then the inverse is a function. Example Consider f(x) = x 4. a. Is the inverse of f(x) a function? Explain. b. Find f -1 (x). Graph f(x) and f -1 (x). Example 3 The fixed costs for manufacturing a particular stereo system are $96,000 and the variable costs are $80 per unit. a. Write and equation that expresses the total cost C(x) as a function of x given that x units are manufactured. b. Determine the equation for the inverse process and describe the real-world situation it models. c. Determine the number of units that can be made for $144,000. 3

4 Example 4 Given f(x) = 3x + 7 and g x x 7 3, determine whether f(x) and g(x) are inverses of each other. Section 5 Continuity and End Behavior Discontinuous Function Cannot be traced without lifting your pencil Infinite Discontinuity Jump Discontinuity Point Discontinuity f x become greater and greater as the graph approaches a given x-value (a vertical asymptote) The graph stops at a given x-value and begins again at a different y-value An x-value for which the function is undefined, but the pieces of the graph match up (typically when a factor cancels from the equation). Continuity Test A function is continuous at x = c if it satisfies the following conditions: (1) the function is defined at c, f(c) exists () the function approaches the same y-value on the left and right of c (3) the y-value that the function approaches from each side is f(c) 4

5 Example 1 Determine whether each function is continuous at the given x-value. x 4 x a. y = 3x + x 7; x = 1 b. f x ; x c. f x x x if x 1 if 4 x A function may have a discontinuity at one or more x-values but be continuous on an interval of other x-values. For example: Continuity on an Interval A function is continuous on an interval, if and only if it is continuous at each number x in in the interval. Example A shipping company offers insurance for its express delivery service. For a package valued at no more than $500, insurance is included in the $1.00 fee. For $ to $5000, it costs an additional $0.95 per $100 value. The graph summarizes the cost of express mail insurance. a. Use the continuity test to show that the step function is discontinuous. b. Explain why a continuous function would not be appropriate to model express delivery rates. End Behavior Describes what the y-values do as x and x. Example 3 Describe the end behavior of the functions. a. f(x) = 5x 3 b. g(x) = -x 9 5

6 Increasing Function Decreasing Function A function is increasing on an interval I, if and only if, for every a and b contained in I, f(a) < f(b) whenever a < b. A function is decreasing on an interval I, if and only if, for every a and b contained in I, f(a) > f(b) whenever a > b. Constant Function A function is constant on an interval I, if and only if, for every a and b contained in I, f(a) = f(b) whenever a < b. Example 4 Graph each function (in your calculator). Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing. (Use interval notation.) a. f(x) = x 7 increasing decreasing b. f x 1 x increasing decreasing c. h(x) = 5x 3 + x x + 4 increasing decreasing 6

7 Section 6 Critical Points and Extrema Critical points Maximum Minimum Point of Inflection Those points on a graph at which a line drawn tangent to the curve is horizontal or vertical. The graph is increasing to the left of x = c and is decreasing to the right The graph is decreasing to the left of x = x and increasing to the right. The graph changes curvature. Absolute maximum The greatest value a function assumes over its domain Absolute minimum The lease value a function assumes over its domain Extremum General terms for maximum(s) and minimum(s) Relative maximum The greatest value on some interval of f(x) Relative minimum The least value on some interval of f(x) Example 1 Locate the extrema for the graph of y = f(x). Name and classify the extrema of the function. 7

8 Example Locate the extrema for the graph of y = f(x). Name and classify the extrema of the function. Example 3 Use a graphing calculator to graph f(x) = x 3 8x + 3 and to determine and classify its extrema. Example 4 One hour after x milligrams of a particular drug are given to a person, the rise in body temperature T(x), in degrees x Fahrenheit, is given by T x x. The model has a critical point at x = 4.5. Determine if this critical point is a 9 maximum. Why should a doctor be aware of this critical point? 8

9 Section 7 Graphs of Rational Functions Rational Function g x The quotient of two polynomials f ( x), h x 0 h x Asymptotes A line that the graph approaches, but never crosses Vertical Asymptote x = a if f(x) or f(x) - as x a from the right or left Horizontal Asymptote y = b if f(x) b as x or x - At the value(s) of x that make the denominator equal zero there are two different types of discontinuities, removable and essential. Example: Removable Discontinuity Yields a hole in the graph x x 8 ( x) x 4 f Example: Essential Discontinuity Yields a vertical asymptote 3 g x 4 x 9

10 To determine horizontal asymptotes in a rational function, look at the leading terms in the numerator and denominator. If the degree in the numerator is the same as the If the degree in the numerator is greater than the degree of the denominator, divide the leading terms to degree of the denominator, then there is no horizontal determine the equation for the horizontal asymptote. asymptote and the end behavior is the same as the end behavior of the quotient of the leading terms. 3x 1 f x x x 3 4x x 6 f x x 3 Special case: If the degree in the numerator is exactly one greater than that of the denominator, then there is a slant asymptote. The equation of the slant asymptote can be obtained by using polynomial long division to divide the numerator by the denominator. 4x 6 37 x f x x 4 If the degree of the numerator is less than the degree of the denominator, then y = 0 is a horizontal asymptote. f x 7 x 5 10