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MAC1106 Lecture Notes - Lecture 6: Greatest Common Divisor

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10 Jan 2018

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Miami Dade College

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Mathematics: Calculus & Pre-Calculus

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Provide a generalization to the terms listed in this section. What are intercepts: intercepts are points on a graph where the points touch or cross the coordinate axes, there are three di erent types of intercepts: X-intercepts are points on a graph where the points cross or touch the x-axis. Y -intercepts are points on a graph where the points cross or touch the y-axis. The origin is the only point on a graph where the point crosses or touches both the x-axis and y-axis. How do you nd the intercepts: x-intercepts. Substitute 0 for y, which is f (x). Solve for the x variable: y -intercepts. Intercepts example f (x) = x2 3x + 4. Y intercept(s) y = 02 3(0) + 4 y = 0 0 + 4 (0, 4) (x + 4)(x 1) = 0 y = 4 x = 4, 1 ( 4, 0) , (1, 0) (cid:0) .

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Algebra

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Algebra

Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = x2 -6x + 8 Complete the square by entering the correct numbers into the expression below. f(x) = (x-( ))2 + ( ) Choose the correct graph below. The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The intercept(s) is/are . (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of the function below. The range of the function f is (Type your answer in interval notation.) Is f continuous on its domain? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Yes, f is continuous on its domain. No, f is discontinuous at x = . (Use a comma to separate answers as needed.) The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of f(x) below. The range of the function f is (Type your answer in interval notation.) Is f continuous on its domain? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. No, f is discontinuous at x = |. (Use a comma to separate answers as needed.) Yes, f is continuous on its domain. Find the distance d(P1 P2) between the given points P1 and P2. P1 =(-5,2) P2 = (4.4) d(P1, P2) = (Simplify your answer. Type an exact answer, using radicals as needed.) Solve the inequality. Express your answer using set notation or interval notation. Graph the solution set. Choose the correct answer below that is the solution set to the inequality. Choose the correct graph below that is the solution set to the inequality. Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry' with respect to the x-axis, y-axis, or the origin. Is the graph that of a function? Yes No If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and fill in any answer boxes within your choice. The domain is The range is (Type your answers in interval notation.) The graph is not a function. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. The graph of an equation is given. List the intercepts of the graph. Based the graph, tell whether the graph is symmetric with the respect to the x-axis, y-axis, and/or the origin What are the intercept(s) of the graph? Select the correct choice below and fill in any answer boxes within your choice. The intercept(s) of the graph are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Determine if the graph is symmetric. It is symmetric with respect to the x-axis. It is symmetric with respect to the y-axis. It is symmetric with respect to the origin. The graph is not symmetric. Solve the following inequality. Graph the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution set is (Type your answer in interval notation.) There is no real solution. Graph the solution set. Choose the correct graph below. Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry with respect to the x-axis, y-axis, or the origin. Is the graph that of a function? Yes No What are the domain and range of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The domain is . The range is . (Type your answers in interval notation.) The graph is not that of a function. What are the intercepts of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Find the real solutions, it any. ot the following equation. Use the quadratic formula. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. The solution set is . (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) The solution is not a real number.

Show transcribed image text Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = x2 -6x + 8 Complete the square by entering the correct numbers into the expression below. f(x) = (x-( ))2 + ( ) Choose the correct graph below. The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The intercept(s) is/are . (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of the function below. The range of the function f is (Type your answer in interval notation.) Is f continuous on its domain? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Yes, f is continuous on its domain. No, f is discontinuous at x = . (Use a comma to separate answers as needed.) The function f is defined as follows. Find the domain of the function. Locate any intercepts. Graph the function. Based on the graph, find the range. Is f continuous on its domain? The domain of the function f is (Type your answer in interval notation.) Locate any intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Choose the correct graph of f(x) below. The range of the function f is (Type your answer in interval notation.) Is f continuous on its domain? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. No, f is discontinuous at x = |. (Use a comma to separate answers as needed.) Yes, f is continuous on its domain. Find the distance d(P1 P2) between the given points P1 and P2. P1 =(-5,2) P2 = (4.4) d(P1, P2) = (Simplify your answer. Type an exact answer, using radicals as needed.) Solve the inequality. Express your answer using set notation or interval notation. Graph the solution set. Choose the correct answer below that is the solution set to the inequality. Choose the correct graph below that is the solution set to the inequality. Determine whether the graph below is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry' with respect to the x-axis, y-axis, or the origin. Is the graph that of a function? Yes No If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and fill in any answer boxes within your choice. The domain is The range is (Type your answers in interval notation.) The graph is not a function. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. The graph of an equation is given. List the intercepts of the graph. Based the graph, tell whether the graph is symmetric with the respect to the x-axis, y-axis, and/or the origin What are the intercept(s) of the graph? Select the correct choice below and fill in any answer boxes within your choice. The intercept(s) of the graph are (Type an ordered pair. Use a comma to separate answers as needed.) There are no intercepts. Determine if the graph is symmetric. It is symmetric with respect to the x-axis. It is symmetric with respect to the y-axis. It is symmetric with respect to the origin. The graph is not symmetric. Solve the following inequality. Graph the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution set is (Type your answer in interval notation.) There is no real solution. Graph the solution set. Choose the correct graph below. Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find its domain and range. the intercepts, if any. any symmetry with respect to the x-axis, y-axis, or the origin. Is the graph that of a function? Yes No What are the domain and range of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The domain is . The range is . (Type your answers in interval notation.) The graph is not that of a function. What are the intercepts of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Find the real solutions, it any. ot the following equation. Use the quadratic formula. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. The solution set is . (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) The solution is not a real number.

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Sketch using these guidelines

pul aill al thes information together to sketch graphs that reveal the important You might ask: Why don't we just use a graphing calculator or computer to graph a It's true that modern technology is capable of producing very accurate graphs. But K 21+1+2 features of functions curve? Why do we need to use calculus? even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on tech- nology. (See "Graphing Calculators and Computers" at www.stewartcalcelus.com, espe 4 discover the most interesting aspects of graphs and in many cases to and minimum points and inflection points exocily instead of approximately cially Examples 1, 3.4, and 5. See also Section 4.6.) The use of calculus enables us to calculate maximum FIGURE 1 For instance, Figure I shows the gr aph of /(x)-è¨-21x, + 18x + 2.At first glance it seems reasonable: It has the same shape as cubic curves like y x and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x-0.75 and a minimum when x 1. Indeed if we zoom in to this portion of the graph, we see that behavior exhibited in Pigure 2 Without calculus, we could easily have overlooked it. 6a, FIGURE 2 In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the foilowing information. We don't assume that you have a graphing device, but if you do have one you should use it as a check on your work. â Guidelines for Sketching a Curve every item is relevant to every function. (For instance, a to make a sketch that displays the most important aspects of the function. The following checklist is intended as a guide to sketching a curve y- f(x) by hand. Not given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need A. Domain It's often useful to start by determining the domain D of f, that is, the set B. Intercepts The y-intercept is f(o) and this tells us where the curve intersects the of values of x for which f(x) is defined y-axis. To find the x-intercepts, we set y-0and solve for xã(You can omit this step if the equation is difficult to solve.) ry (a) Even fusction: reflectional symmetry all x in D, that is, the equation of the curve is unchanged when x is replacea by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for 0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a). Here are some examples: y-ey-ry x). and for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180Â° about the origin; see Figure 3(b).] Some b) Oud function rotational symmetry FIGURE 3 simple examples of odd functions are y-xy-x'y - and y-sin