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Understanding Functions Through Graphs: A Complete Guide

Learn how to extract valuable information from a graph of a function, including identifying increasing/decreasing values, maximum/minimum points, domain and range. Insights on interpreting graphs to analyze functions effectively.

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Understanding Functions Through Graphs: A Complete Guide

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  1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

  2. Functions 3

  3. Getting Information from the Graph of a Function 3.3

  4. The Graph of a Function • Many properties of a function are more easily obtained from a graph than from the rule that describes the function. • We will see in this section how a graph tells us: • Whether the values of a function are increasing or decreasing • Where the maximum and minimum values of a function are.

  5. Values of a Function;Domain and Range

  6. The Values of a Function • A complete graph of a function contains all the information about a function. • This is because the graph tells us which input values correspond to which output values. • To analyze the graph of a function, we must keep in mind that the height of the graph is the value of the function. • So we can read off the values of a function from its graph.

  7. E.g. 1—Find the Values of a Function from a Graph • The function T graphed here gives the temperature between noon and 6 P.M. at a certain weather station.

  8. E.g. 1—Find the Values of a Function from a Graph (a) Find T(1), T(3), and T(5). • Which is larger, T(2) or T(4)? • Find the value(s) of x for which T(x) = 25. • Find the value(s) of x for which T(x) ≥ 25.

  9. Example (a) E.g. 1—Find Values of a Function • T(1) is the temperature at 1:00 P.M. • It is represented by the height of the graph above the x-axis at x = 1. • Thus, T(1) = 25. • Similarly, T(3) = 30 and T(5) = 20.

  10. Example (b) E.g. 1—Find Values of a Function • The graph is higher at x = 2 than at x = 4. • It follows that T(2) is larger than T(4).

  11. Example (c) E.g. 1—Find Values of a Function • The height of the graph is 25 when x is 1 and when x is 4. • In other words,the temperatureis 25 at 1:00 P.M.and 4:00 P.M.

  12. Example (d) E.g. 1—Find Values of a Function • The graph is higher than 25 for x between 1 and 4. • In other words,the temperaturewas 25 or greaterbetween 1:00 P.M.and 4:00 P.M.

  13. Finding the Domain and Range from a Graph • The graph of a function helps us to picture the domain and range of the function on the x-axis and y-axis, as shown.

  14. E.g. 2—Finding the Domain and Range from a Graph • Use a graphing calculator to draw the graph of . • Find the domain and range of f.

  15. E.g. 2—Finding the Domain and Range from a Graph • (a) The graph is shown. • (b) We see that: • The domain is [–2, 2]. • The range is [0, 2].

  16. Increasing and Decreasing Functions

  17. Increasing and Decreasing Functions • It is very useful to know where the graph of a function rises and where it falls.

  18. Increasing and Decreasing Functions • The graph shown here rises, falls, then rises again as we move from left to right. • It rises from A to B, falls from B to C, and rises again from C to D.

  19. Increasing and Decreasing Functions • The function f is said to be: • Increasing when its graph rises. • Decreasing when its graph falls.

  20. Increasing Function—Definition • f is increasing on an interval Iif f(x1) < f(x2) whenever x1 < x2 in I.

  21. Decreasing Function—Definition • f is decreasing on an interval Iif f(x1) > f(x2) whenever x1 < x2 in I.

  22. E.g. 3—Intervals Where a Func. Increases & Decreases • The graph gives the weight W of a person at age x. • Determine the intervals on which the function W is increasing and on which it is decreasing.

  23. E.g. 3—Intervals Where a Func. Increases & Decreases • The function is: • Increasing on [0, 25] and [35, 40]. • Decreasing on [40, 50]. • Constant (neither increasing nor decreasing) on [25, 35] and [50, 80].

  24. E.g. 3—Intervals Where a Func. Increases & Decreases • This means that: • The person gained weight until age 25. • He gained weight again between ages 35 and 40. • He lost weight between ages 40 and 50.

  25. E.g. 4—Finding Where a Function Incrs. & Decrs. • (a) Sketch the graph of the functionf(x) = 12x2 + 4x3 – 3x4 • (b) Find the domain and range of f. • (c) Find the intervals on which f increases and decreases.

  26. Example (a) E.g. 4—Sketching the Graph • We use a graphing calculator to sketch the graph here.

  27. Example (b) E.g. 4—Domain • The domain of f is because f is defined for all real numbers.

  28. Example (b) E.g. 4—Range • Using the TRACE feature on the calculator, we find that the highest value is f(2) = 32. • So, the range off is (–∞, 32].

  29. Example (c) E.g. 4—Increase & Decrease • We see that f is: • Increasing on (–∞, 1] and [0, 2]. • Decreasing on[–1, 0] and [2, ∞).

  30. E.g. 5—Finding Where a Function Incrs. & Decrs. • (a) Sketch the graph of the function f(x) = x2/3. • (b) Find the domain and range of the function. • (c) Find the intervals on which f increases and decreases.

  31. Example (a) E.g. 5—Sketching the Graph • We use a graphing calculator to sketch the graph here.

  32. Example (b) E.g. 5—Domain & Range • From the graph, we observe that: • The domain of f is . • The range is [0, ∞).

  33. Example (c) E.g. 5—Increase & Decrease • We see that f is: • Decreasing on (-∞, 0]. • Increasing on[0, ∞).

  34. Local Maximum and MinimumValues of a Function

  35. Local Maximums and Minimums of a Function • Finding the largest or smallest values of a function is important in many applications. • For example, if a function represents revenue or profit, then we are interested in its maximum value. • For a function that represents cost, we would want to find its minimum value. • See Focus on Modeling: Modeling with Functions on pages 280-289 for many such examples.

  36. Local Maximums and Minimums of a Function • We can easily find these values from the graph of a function. • We first define what we mean by a local maximum or minimum.

  37. Local Maximums of a Function—Definition • The function value f(a) is a local maximum value of f if • f(a) ≥ f(x) when x is near a • This means thatf(a) ≥ f(x) for all xin some open intervalcontaining a. • In this case, we saythat f has a localmaximum at x = a.

  38. Local Minimums of a Function—Definition • The function value f(a) is a local minimum value of f if • f(a) ≤ f(x) when x is near a • This means thatf(a) ≤f(x) for all xin some open intervalcontaining a. • In this case, we saythat f has a localminimum at x = a.

  39. Using Graphing Devices to Find Extreme Values • Suppose there is a viewing rectangle such that the point (a, f(a)) is the highest point on the graph of f withinthe viewing rectangle (not on the edge).

  40. Local Maximum Value • Then, the number f(a) is called a local maximum valueof f. • Notice that f(a) ≥ f(x) for all numbers x that are close to a.

  41. Using Graphing Devices to Find Extreme Values • Similarly, suppose there is a viewing rectangle such that the point (b, f(b)) is the lowest point on the graph of f within the rectangle.

  42. Local Minimum Value • Then, the number f(b) is called a local minimum valueof f. • In this case, f(b) ≤ f(x) for all numbers x that are close to b.

  43. E.g. 6—Finding Local Maxima & Minima from a Graph • Find the local maximum and minimum values of the function f(x) = x3 – 8x + 1 correct to three decimals.

  44. E.g. 6—Local Maxima & Minima from a Graph • Looking at the graph of f, there appears to be: • One local maximum between x = –2 and x = –1. • One local minimum between x = 1 and x = 2. • Let’s find the coordinates of the local maximum point first.

  45. E.g. 6—Local Maxima & Minima from a Graph • We zoom in to enlarge the area near this point. • Using the TRACE feature on the graphing device, we move the cursor along the curve and observe how the y-coordinates change.

  46. E.g. 6—Local Maxima & Minima from a Graph • The local maximum value of y is: 9.709 • This value occurs when x is –1.633, correct to three decimals.

  47. E.g. 6—Local Maxima & Minima from a Graph • By zooming in to the viewing rectangle shown, we find that the local minimum value is about –7.709. • This value occurs when x ≈ 1.633.

  48. Using Graphing Devices to Find Extreme Values • The maximum and minimum commands on a TI-82 or TI-83 calculator provide another method for finding extreme values of functions. • We use this method in the next example.

  49. E.g. 7—A Model for the Food Price Index • A model for the food price index (the price of a representative “basket” of foods) between 1990 and 2000 is given by the functionwhere: • t is measured in years since midyear 1990, so 0 ≤t ≤ 10. • I(t) is scaled so that I(3) = 100.

  50. E.g. 7—A Model for the Food Price Index • Estimate the time when food was most expensive during the period 1990–2000.

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