September 08, Graph y 2 =x. How? Is it a function? Function?

Transcription

1 Graph y 2 =x How? Is it a function? Function?

2 Section 1.3 Graphs of Functions Objective: Analyze the graphs of functions. Important Vocabulary Graph of a function The collection of ordered pairs ( x, f(x)) such that x is in the domain of f. Graph same as equations in two variables since y=f(x). Greatest integer function The greatest integer function is denoted by [ x ] and is defined as the greatest integer less than or equal to x. Use "int(" to graph Step function A function whose graph resembles a set of stair steps. Example of... Even function A function y = f(x) is even if, for each x in the domain of f, f( x) = f(x). Odd function A function y = f(x) is odd if, for each x in the domain of f, f( x) = f(x). I. The Graph of a Function Explain the use of open or closed dots in the graphs of functions. The use of dots, open or closed, at the extreme left or right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points. To find the domain of a function from its graph,... examine the graph to see for which x-values the graph is plotted. To find the range of a function from its graph,... examine the graph to see for which y-values the graph is plotted. The Vertical Line Test for functions states... that a set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

3 Functions? II. Increasing and Decreasing Functions Graph f(x)=x 3 +3x 2 A function f is increasing on an interval if, for any x 1 and x 2 in the interval,... x 1 < x 2 implies f(x 1 ) < f(x 2 ). A function f is decreasing on an interval if, for any x 1 and x 2 in the interval,... x 1 < x 2 implies f(x 1 ) > f(x 2 ). A function f is constant on an interval if, for any x 1 and x 2 in the interval,... f(x 1 ) = f(x 2 ). Given a graph of a function, to find an interval on which the function is increasing... search for an interval over which the graph rises from left to right. Given a graph of a function, to find an interval on which the function is decreasing... search for an interval over which the graph falls from left to right. Given a graph of a function, to find an interval on which the function is constant... search for an interval over which the graph is level.

4 III. Relative Minimum and Maximum Values A function value f(a) is called a relative minimum of f if... there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). A function value f(a) is called a relative maximum of f if... there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) f(x). The point at which a function changes from increasing to decreasing is a relative maximum. The point at which a function changes from decreasing to increasing is a relative minimum. To approximate the relative minimum or maximum of a function using a graphing utility,... use the zoom and trace features to approximate the lowest or highest point on the graph, OR use a built-in program or feature for finding minimum or maximum values. Ex: Suppose a function C represents the annual number of cases (in millions) of chicken pox reported for the year x in the United States from 1960 through Interpret the meaning of the function s minimum at (1998, 3). The lowest number of cases of chicken pox reported during this period was 3 million cases in Ex: The profit for a new company can be modeled by P=0.225x x x Graph the function and use the maximum command to find the relative max. Should be approximately $1,822,680.

5 IV. Graphing Step Functions and Piecewise-Defined Functions The Greatest Integer Function The graph of the greatest integer function jumps vertically one unit at each integer and is constant (a horizontal line segment) between each pair of consecutive integers. Ex: Let f(x)= [ x ], the greatest integer function. Find f(3.74). Graph MATH: NUM 5: int( Note: change MODE from Connected to Dot and ZOOM in Answer: 3 Piecewise-Defined Function sketch the graph of each equation of the piecewise-defined function over the appropriate portion of the domain. Ex: Sketch the graph of f(x)= x+1, if x < 2 1- x, if x > 2 f(x)= x+1, if x < 2 1- x, if x > 2

6 V. Even and Odd Functions A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph, (- x, y) is also on the graph. A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, - y) is also on the graph. A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph, (- x, - y) is also on the graph. A function whose graph is symmetric with respect to the y-axis is a(n) A function whose graph is symmetric with respect to the origin is a(n) even odd function. function. The graph of a (nonzero) function cannot be symmetric with respect to the x-axis. So... even ---> f(-x)=f(x) odd ---> f(-x)=-f(x) Even or odd? a. b. c.

7 Assignment: p. 38 (2-26e, 32, 33, 41-43, 48, 50, 54, 66, 68, 82, 87, 89, 90) September 08, 2014