Contents: This page corresponds to § 1.4 (p. 116) of the text. Show
Suggested Problems from Text p. 124 #1, 2, 5, 8, 9, 11, 16, 17, 21, 25, 27, 29, 31, 39, 40, 47, 50, 51, 52, 54, 57, 64, 65, 66
Defining the Graph of a FunctionThe graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation. Example 1.
Exercise 1:
Example 2.
Exercise 2:
We have seen that some equations in x and y do not describe y as a function of x. The algebraic way see if an equation determines y as a function of x is to solve for y. If there is not a unique solution, then y is not a function of x. Suppose that we are given the graph of the equation. There is an easy way to see if this equation describes y as a function of x. Return to Contents Vertical Line Test
Example 3.
Return to Contents Characteristics of GraphsConsider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1 as the Slope-Intercept form of the equation of a line with slope 2 and y-intercept (0,1). Think of a point moving on the graph of f. As the point moves toward the right it rises. This is what it means for a function to be increasing. Your text has a more precise definition, but this is the basic idea. The function f above is increasing everywhere. In general, there are intervals where a function is increasing and intervals where it is decreasing. The function graphed above is decreasing for x between -3 and 2. It is increasing for x less than -3 and for x greater than 2.
Using interval notation, we say that the function is
Exercise 3:
Some of the most characteristics of a function are its Relative Extreme Values. Points on the functions graph corresponding to relative extreme values are turning points, or points where the function changes from decreasing to increasing or vice versa. Let f be the function whose graph is drawn below. f is decreasing on (-infinity, a) and increasing on (a, b), so the point (a, f(a)) is a turning point of the graph. f(a) is called a relative minimum of f. Note that f(a) is not the smallest function value, f(c) is. However, if we consider only the portion of the graph in the circle above a, then f(a) is the smallest second coordinate. Look at the circle on the graph above b. While f(b) is not the largest function value (this function does not have a largest value), if we look only at the portion of the graph in the circle, then the point (b, f(b)) is above all the other points. So, f(b) is a relative maximum of f. f(c) is another relative minimum of f. Indeed, f(c) is the absolute minimum of f, but it is also one of the relative minima. Here again we are giving definitions that appeal to your geometric intuition. The precise definitions are given in your text. Return to Contents Approximating Relative ExtremaFinding the exact location of a function's relative extrema generally requires calculus. However, graphing utilities such as the Java Grapher may be used to approximate these numbers. Note on terminology:
Example 4.
Exercise 4:
Return to Contents Even and Odd FunctionsA function f is even if its graph is symmetric with respect to the y-axis. This criterion can be stated algebraically as follows: f is even if f(-x) = f(x) for all x in the domain of f. For example, if you evaluate f at 3 and at -3, then you will get the same value if f is even. This condition is very easy to check with the Java Grapher. Example 5.
A function f is odd if its graph is symmetric with respect to the origin. This criterion can be stated algebraically as follows: f is odd if f(-x) = -f(x) for all x in the domain of f. For example, if you evaluate f at 3, you get the negative of f(-3) when f is odd. If you enter any function in the f box of the Grapher and enter -f(-x) in the g box, then the graph of g is the reflection through the origin of the graph of f. So, if f is not odd, then you see two distinct graphs. If f is odd, you see only one graph. Example 6.
Return to Contents Practice QuizHow do you find a function on a graph?How To: Given a graph, use the vertical line test to determine if the graph represents a function. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. If there is any such line, the graph does not represent a function.
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