Functions 1. Any set of ordered pairs is a relation. Graph each set of ordered pairs. A. x y B. x y C. x y -3-6 -2-1 -3 6-1 -3-3 0-2 4 0-1 6 3-1 1 3 5-2 1 0 0 5 6-1 4 1 1 A. B. C. 2. A function is a special type of relation that assigns exactly one value of y for each value of x. Circle the pairs above that have the same value for x and different values of y. Which of the relations above are functions? 3. Finish filling in the tables using the given equations and then plot the points and connect them. A. y = 2x 1 x = y 2 y = x 2 x y B. x y C. x y -3-7 9-3 -3-2 -5 4-2 -1 1-1 0 0 0 1 1 1 2 4 2 3 9 3
A. B. C. 4. For each graph, are there any values of x for which there are more than one value for y? A B C. 5. Perform the Vertical Line Test, that is, roll your pencil vertically across each graph left to right. Does it touch any of the lines in more than one place? A B C. 6. A and C are functions. B is not a function. From the above questions, what can you determine is true about the graph of a function. 7. Why does the Vertical Line Test work when checking if a given graph represents a function? 8. Which of the following graphs represent functions? A. B. C.
D. E. F. A. B. C. D. E. F.
Domain and Range 1. The domain of a function described by this set of ordered pairs in (x, y)-form: {(-2, 4), (-1, 1 ), (0, 0), (1, 1). (2, 4)} is {-2, -1, 0, 1, 2} What variable does the domain represent? 2. Find the domain of the function described by this set of ordered pairs: {(-3, 6), (-1, 2), (5, -10), (0, 0), (2, -4), (3, -6), (4, -8)} 3. The range for the ordered pairs in #1 is {4, 1, 0, 1, 4}. What variable does the range represent? 4. Find the range of the function described by the set of ordered pairs in #2. 5. The domain is the set of values assigned to x, the range is the set of corresponding values of y. Find the domain and range for the following set of ordered pairs: {(2, 4), (5, 9), (a, b), (-6, 4), (7, -12), (-p, q), (0, 0)} Domain: Range: 6. Look at the following graph: Is there a point that corresponds to x = -1? Is -1 in the domain of the graph of this function? Is there a point that corresponds to x = 5? Is 5 in the domain of the graph of this function? For a graph of a function, the domain is any value of x for which there is a corresponding value of y, hence, a point on the graph. The domain for the function represented by this graph is all real numbers > 1.
7. Refer to the graph in #6. Is there any point that corresponds to y = -2? Is -2 in the range of the graph of this function? Is there a point that corresponds to y = 10? Is 10 in the range of the graph of this function? What is the range of the function represented by the graph? 8. Find the domain and range of the functions described by the following graphs. A B Domain A = Domain B = Range A = Range B = C D Domain C = Domain D = Range C = Range D =
E F Domain E = Domain F = Range E = Range F = G H Domain G = Domain H = Range G = Range H = When a function is given as a rule, y is a function of x and is written as y = f(x). The variable x is called the independent variable because you can choose the value of x you want to evaluate. The domain is the set of permissible values of x. The range is the set of all the possible values you can get out of the function. We call y the dependent variable because it depends on the value of x that you choose. 9. Graph the functions, then state the domain and range for each. a) y = x sinx Range = Domain = b) y = 4sinx Range = Domain =
10. For many functions, the domain is the set of all real numbers but sometimes there are values for x that won t work. Determine which values of x are not in the domain of the following functions. a) y = 1/x b) y = 5/(x +3) c) y = (3+x)/x 2 d) y = 2 x 9 x 3 e) y = x +3 f) y = 2 x g) y = 5x 12 h) y = 3 x 11. What arithmetic principle did you use to answer a, b, c, and d? 12. What arithmetic principle did you use to answer e, f, g, and h? The domain of a function is the set of all real numbers except for the values of x that do not give real solutions. 13. Graph the function in #10 on your calculator. What is the range for each of the functions? a) g) b) h) c) d) e) f)
Zeros (Roots) of Polynomials 1. Using your calculator, draw the graphs for the following functions. At what point(s) do the graphs cross the x-axis? a) y = 2x + 4 b) y = 4x - 3 c) y = x 2-4 d) y = 2x 2 + 3x - 6 e) y = 3x 3-5x 2-26x 8 f) y = 4x 3-12x 2 + 3x + 5 2. What do the coordinates of these points have in common? The x-coordinates of the points where the graph crosses the x-axis are called the zeroes or real roots of the function. 3. For each of the functions from #1, factor the right hand side of the expression (e and f are done for you!) and solve the equations for y = 0. a) y = 2x + 4 b) y = 4x 3 c) y = x 2 4 d) y = 2x 2 + 3x 6 e) y = 3x 3-5x 2-26x 8 y = (3x + 1)(x - 4)(x + 2) f) y = 4x 3-12x 2 + 3x + 5 y = (2x + 1)(x 1)(2x 5) 4. What do you notice about the solutions in #3 and the points you found in #1? 5. If the polynomials are factored, how can you guess the zeroes without graphing?
6. Look at your answers in #1. How many times did each graph cross the x-axis? Do you see any relationship between the degree of the polynomial function and the number of times the graph crosses the x-axis? a) b) c) d) e) f) 7. Write an equation for a polynomial function which has zeroes of -2 and 5. Think about what degree this polynomial will be. Enter your equation into your calculator to check your answer graphically. 8. Write a polynomial function of least degree which has zeroes at -1, 2, and 5. What degree will this polynomial be? Enter your equation into your calculator to check your answer graphically. 9. Graph the functions y = (x - 3)(x + 4), y = 3(x - 3)(x + 4), and y = 0.5(x - 3)(x + 4). Do they have different zeroes? What can you conclude about other functions represented by y = k(x - 3)(x + 4), where the constant k represents any real number.. 10. Write two more possible equations for the polynomial with same zeroes described in #7, one with the same degree and another one of different degree.
11. Write two more possible equations for the polynomial with the same zeroes described in #8, one with the same degree and another one of different degree. 12. Write an expression for the function represented by each of the following graphs. Enter your equation into your calculator to check your answer graphically. A B A. B. 13. How many zeroes do the following equations have? Graph each equation to check your answer. y = 12x 2-6x - 6 y = 6x 4 + 5x 3-15x 2 + 4 y = 2x 5 + x 4-39x 3-6x 2 + 80x - 32 y = 6x 3 + 11x 2-4x - 4 14. Consider the function y = x 2-4x + 4. How many zeroes do you think this function could have? Now graph the function. Does it cross the x-axis as many times as you thought it would? Why do you think this happened? When you factor the expression, you get y = (x - 2)(x - 2) = (x 2) 2. There are still two real roots (2 and 2) but they are not distinct numbers. This happens any time the factors are the same. We call x = 2 a double root.
15. The degree of a polynomial tells you the most number of times the graph of a function will cross the x-axis. There could be fewer real roots, but never more. How many times would you expect the graph of y = 2x 2 + 2x + 4 to cross the x-axis? How many times does it cross? How many real roots does this function have? 16. Consider the function y = x 3 + 4x 2 + 3x + 5. How many roots do you think this function could have? How many times does the graph actually cross the x-axis? How many real roots does this function have?
Local Maxima and Minima 1. Look at the graph of the function y = x 2. What is the lowest point or valley you see on the graph? Is there more than one? Does the entire graph have a highest point or peak? Explain your reasoning. 2. Now look at the graph of the function y = -x 2. What is the highest point or peak you see on the graph? Is there more than one? Does the entire graph have a lowest point of valley? Explain your reasoning. What is happening to the values of y as x increases from -5 to 0? What is happening to the values of y as x increases from 0 to 5?
3. Below is a graph of the function y = x *(x - 1)(x + 1) + 2. Look at the point (-0.58, 2.38). Is it the highest point on the entire graph? Is it the highest point in the interval [-2, 0]? If there is a highest point(s) on the graph of a function in an open interval, the point(s) is called the local maxima. What happens to the values of y as x increases in the interval before the local maxima, [-3, -0.58]? What happens to the values of y as x increases in the interval after the local maxima [-0.58, 0]? What can you generalize about the behavior (is it increasing, decreasing?) of the graph immediately before and immediately after the local maxima?
Now look at the point (0.58, 1.62). Is it the lowest point on the entire graph? Is it the lowest point in the interval [0, 1]? If there is a lowest point(s) on the graph of a function in an open interval, the point(s) is called the local minima. What happens to the values of y as x increases in the interval before the local minima, [0, 0.58]? What happens to the values of y as x increases in the interval after the local minima [0.58, 3]? the What can you generalize about the behavior (is it increasing, decreasing?) of graph immediately before and immediately after the local minima? If the values of y increase (get bigger) as the values of x in a given interval increase(going left to right), we say the function is increasing in that interval. If the values of y decrease (get smaller) as the values of x in a given interval get bigger (going left to right), we say the function is decreasing in that interval. 4. For the graph of the function y = x*sinx. Find the local maxima and minima and specify the intervals. Where is the function increasing in the interval [-10, 10]? Where is it decreasing in that interval?
Extension problem: 5. You want to fold a box with a lid from a sheet of cardboard 20 inches by 25 inches. You cut x by x squares from two corners and x by 12.5 inch rectangles from the other two corners as shown in the diagram below. Write the equation for the volume of the box then determine what value of x would give your box the maximum volume V? x --------12.5--------- 20 x x --------------------------- 25 -------------------------