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2.1 Describing Graphs of Functions

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If we examine a typical graph the function y = f(x), we can observe that for an interval throughout which the function is defined, that the function might be increasing, decreasing or neither.

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We say that a function is increasing on an interval if x1 and x2 are in the interval such that x1 < x2 and we have f(x1) < f(x2). Further, we say that f(x) is increasing at x = c provided that f(x) is increasing in some open interval on the x-axis that contains c.

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We say that a function is decreasing on an interval if x1 and x2 are in the interval such that x1 f(x2). Further, we say that f(x) is decreasing at x = c provided that f(x) is decreasing in some open interval on the x-axis that contains c.

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Extreme Points

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A relative extreme point ( relative maximum point or relative minimum point) of a function is a point at which its graph changes from increasing to decreasing or vice versa.

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A relative maximum point is a point at which the graph changes from increasing to decreasing.

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A relative minimum point is a point at which the graph changes from decreasing to increasing.

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The maximum value of a function is the largest value that the function assumes on its domain. The minimum value of a function is the smallest value that the function assumes on its domain.

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Note: Functions might or might not have maximum and/or minimum values.

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If a function has a maximum value or minimum value at the endpoint(s) of its domain, we say that the function has an endpoint extreme value.

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Changing slope

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Consider the next two graphs. Note that the graphs of both are increasing, but there is a difference in how they are increasing. What is the difference?

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Graph I

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Graph II

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We note that the slope of graph I is increasing while the slope of graph II is decreasing. In application, we would say that the debt per capita depicted in graph I is rising at an increasing rate. From graph II, we observe that the population is increasing at a declining rate.

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Concavity

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Concavity has a relationship to the tangent lines of a curve.

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We say that a function is concave up at x = a if there is an open interval on the x-axis containing a throughout which the graph of f(x) lies above its tangent line.

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We say that a function is concave down at x = a if there is an open interval on the x-axis containing a throughout which the graph of f(x) lies below its tangent line.

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An inflection point is a point on the graph of a function at which the function is continuous and the concavity of the graph changes, i.e., goes from concave up to concave down, or concave down to concave up.

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Use the terms defined earlier to describe the graph.

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For x < 3, f(x) is increasing and concave down. Relative maximum at the point x = 3. For 3 < x < 4, f(x) is decreasing and concave down. Inflection point at x = 4. For 4 < x < 5, f(x is decreasing and concave up. Relative minimum at x = 5. For x > 5, f(x) is increasing and concave up.

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Intercepts, Undefined Points and Asymptotes

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We have previously discussed the idea of intercepts. Recall that The x-intercept is a point at which a graph intersects the x-axis. (x,0) The y-intercept is a point at which the graph intersects the y-axis. (0,y)

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Note that a function can have at most one y-intercept. Otherwise, its graph would violate the vertical line test for a function. A function may have 0 or more x-intercepts.

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Recall that some functions are not defined for all values of x. For example, is not defined at x = 0 is not defined for x < 0

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Graphs sometimes straighten out and approach some straight line as x increases (or decreases). Theses straight lines are called asymptotes. Asymptotes of a graph may be horizontal, vertical or diagonal.

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The horizontal asymptotes of a graph may be determined by calculating the limits and If either limit exists, then the value of the limit determines a horizontal asymptote.

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We often expect the graph of a function f(x), at a value x that would result in division by zero, to have a vertical asymptote.

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We now have six categories for describing the graph of a function 1.Intervals in which the finction is increasing or decreasing, relative maximum/minimum points 2.Maximum/minimum values 3.Intervals in which a function is concave up or concave down, inflection points. 4.X-intercepts, y-intercepts 5.Undefined points. 6.Asymptotes

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1 Example 6 Sketch the graph of the function Solution I. Intercepts The x-intercepts occur when the numerator of q(x) is zero i.e. when x=1. The y-intercept.

1 Example 6 Sketch the graph of the function Solution I. Intercepts The x-intercepts occur when the numerator of q(x) is zero i.e. when x=1. The y-intercept.

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