SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties

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1 SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties Definition (Graph Form): A function f can be defined by a graph in the xy-plane. In this case the output can be obtained by drawing vertical line from to intersect the graph at a point. Then, from that point draw a horizontal line to get to a point on the y-axis. The y-coordinate of this point approximates. Example 1: Use the function given by the following graph to complete the table Vertical Line Test (VLT): Only certain types of graphs can be used to define functions. The VLT asserts that a given graph represents a function if every vertical line intersects the graph at one point only, because if a vertical line intersects the graph at two (or more) points, then these two (or more ) points have the same input value with different output values,,... and this disqualifies the graph from being a function.

2 Example 2: Which of the following graphs represent a function? 10

3 Definition 2: To sketch the graph of, use a table of values and obtain several ordered pairs. Then plot the pairs as points and connect them with a reasonable curve. The more ordered pairs you obtain the more accurate the graph will be. Note 1: In general this technique is not an efficient way of graphing a function. However, if prior knowledge of the properties of the function is available, then a very accurate graph of the function can be obtained by this method. Otherwise a graphing calculator (or calculus which is beyond the scope of this course) can always be used to produce accurate graphs. Example 3: Sketch the graphs of each function and check with a calculator. a) 11 x y b) x y

4 Finding the Domain and the Range of a (Continuous) Function Graphically: We use examples to demonstrate these concepts. Example 4: Use the graphs of the functions below to find the domain and range. a) Domain: Range: 12 b) Use the graph of the function below to find the domain and range. i) Domain: ii) Range: iii) Estimate values of x for which c) Domain: Range: Definition 3: The y-intercept of a function is the point where the graph of the function crosses the y-axis. So the x-coordinate of the y-intercept is is zero. Hence, to find the y-intercept, set in. This means that the y-intercept [some authors consider as the y- intercept.] A function has at most one y-intercept (why?) Example 5: Find the y-intercept of functions a) b)

5 13 Zeros and x-intercepts of Functions Definition 4: A zero of a function is f is an x-value for which either a real number or a complex number.. A zero of a function is A real zero of a function is an x-intercept of the function, i.e, a point on x-axis where the graph of the function crosses (or is tangent to) the x-axis. We will not discuss complex zeros Hence to find x-intercept(s), set and find the real solutions. Example 6: Estimate the x-intercepts and the y-intercept of the following function (graph) Example 7: Find (without a calculator) the x-intercept(s) and the y-intercept of the functions below. a) b)

6 Remark 1: If solving equation turns out to be difficult, then we utilize a graphing calculator to estimate the real zeros (x-intercept(s).) Examples 8: Use a calculator to estimate the x-intercepts of. Also find the y-intercept. 14 Local Maxima and Local Minima of a Function: A function has a local maximum of at if the point is locally the highest point. A function has a local minimum of at if the point is locally the lowest point. In this course, we will only use a graphing calculator to approximate the local extrema of a function. Example 9: Identify the local maximums and local minimums points of the function given by the graph below

7 Examples 10: Use a graphing calculator to sketch the graph of the functions below, then find all of its local extrema. a) b) 15 Local minimum: Local maximum: Local minimum: Local maximum: Increasing, Decreasing and Constant Functions: A function is called increasing if the y-value gets larger when the x-value gets larger. This means that the graph of the function rises when you look at it from left to right. In notation,. A function is called decreasing if the y-value gets smaller when the x-value gets larger. This means that the graph of the function falls as you look at it from left to right. In notation, If is neither increasing nor deceasing, then is constant..

8 Example 11: For each of the functions given by the graphs below, find the intervals on x-axis, where the function is increasing, decreasing, or constant. 16 Inc. on: Inc. on: Inc. on: Dec. on: Dec. on: Dec.on: Const. on Const. on Const. on Inc. On: Dec. on: Const. on Inc. on: Dec. on: Const. on

9 Example 12: For each of the functions given below, use the coordinates of the local extrema (you found them in example 10) to find the intervals on x-axis, on which the function is increasing, decreasing, or constant. a) 17 Inc. on: Dec. on: Const. on b) Inc. on: Dec. on: Const. on c) Inc. on: Dec. on: Const. on Even Functions: A function is called even if its graph is symmetric with respect to the y-axis. In function notation this means This also means that we change to, the corresponding y-values will not change. So if point is on the graph of, so is point.. Odd Functions: A function In function natation this means is called odd if its graph is symmetric with respect to the origin.. This also means if we change to, then will change to. So if point is on the graph of, so is point.

10 Example 13: Determine (with and without a calculator) whether the function is even, odd, or neither. a) b) 18 c) d) e) f) Application of the Local Extrema Example 14: Use graphing techniques to estimate the value of x for which the volume of the following rectangular box is maximum? Find that maximum volumne.

11 19 Exercise 1. Sketch, on the same xy-plane, the graphs of the following functions with and without a calculator., 2. Approximate the x-intercepts and one y-intercept of the function given below: 3. Find the x-intercept(s) and the y-intercept of each function without a calculator. a) b) 4. Use the graph of to find all x-intercepts. Also find the y- intercept. 5. Use the graph for function f given on the right to find a) The value of f ( 1) b) The x-value such that f (x) = 1 c) The domain of f d) The range of f is 6. Use the graph of to find Domain: Range: 7. Use a graphing calculator to sketch the graph of the functions below, then find all of its local extrema. a) b) c) Local minimum: Local minimum: Local minimum Local Maximum: Local Maximum: Local Maximum

12 8. For each of the functions in problem # 7, use the coordinates of the local extrema to find the intervals on x-axis, on which the function is increasing, decreasing, or constant. a) b) c) Increasing on: Increasing on: Increasing on: Decreasing on : Decreasing on: Decreasing on: 9. Sketch the graph of function. 20 a) Let, then Sketch the graph and discuss its discontinuity. 10 For what value of x would the volume of the rectangular box be maximum? 11 The selling price of a product is given by the (price-demand) function, Where x represents the number of units sold. a) What is the domain of this function? b) Fund a formula for the revenue. c) If the function represents the cost of producing x units of the product, find a formula for the function that represents the profit P. d) What is the loss if the demand is zero (no units are sold)? e) For what demand will the company break even. f) What demand will generate the maximum profit?