# Mathematics Practices

Exploration 15-1a: Cubic Function Graphs

Objective: Learn the shape and some properties of cubic function graphs.

 1. Let Plot the graph using a window with x-range of about [-5, 5] and a y-range of [-30, 30]. Sketch the graph. 2. The graph in Problem 1 crosses the x-axis at three values of x. One of those values is an integer. Which value? Estimate the other two. 3. The integer value in Problem 2 can be used to write a linear factor of . By dividing this factor into , find the other (quadratic) factor. 4. With the help of the quadratic formula , find the exact values of the other two x-intercepts. Confirm that your answers in Problem 2 agree with these exact values. 5. Let Plot the graph of g on your grapher. Sketch on the same axes as in Problem 1. What similarities and differences do you notice in the graphs of f and g? 6. Factor into three linear factors . By setting each factor equal to zero, find the three x-intercepts. What do you notice about two of the x-intercepts? 7. Let Plot the graph of h. Sketch on the same axes as in Problem 1. 8. Find one linear factor of . Use it to find two other values of x that make How do the results agree with the graph? Why do you think values of x that make a polynomial function equal 0 are called zeros of the function rather than just x-intercepts? 9. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-2a: Synthetic Substitution

Objective: Learn how to evaluate f(x) quickly by pencil and paper and how to use
the result to factor f(x).

 1. Let Plot the graph using a window with an x-range of about [-5, 5] and a y-range of [-100, 100]. Sketch the graph here. 2. Find . Mark the corresponding point on your graph. 3. Use long division to divide by See Section 15-2 if you do not recall how to do this. What is the quotient? What is the remainder? 4. What do you notice about the remainder and the value of f (2)? 5. Find f (2) by synthetic substitution. If you don’t recall how to do this, look in Section 15-2. 6. How could you write the answer to Problem 3 directly from the synthetic substitution results ? 7. Find by synthetic substitution. How does the result agree with your graph? 8. Use the result of Problem 7 to find the other two zeros of . Write the complex zeros in terms of i and simplify as much as possible. 9. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-2b: Sum and Product Date:
of the Zeros of a Polynomial

Objective: Discover properties relating the sum and the product of the zeros of a
polynomial function.

 The figure shows the graph of 1. Find and the three zeros of this function. Tell what method you used . 2. Find the product of the zeros, 3. If you factor out 5 from each term in the equation of you get a second factor with leading coefficient equal to 1. How does the product of the zeros you found in Problem 2 relate to the coefficients inside the parentheses ? 4. Find the sum of the zeros, .How does the answer relate to the coefficients in Problem 3? 5. Find the sum of the pairwise products of the zeros, How does the answer relate to the coefficients in Problem 3? 6. Use the patterns you observe in Problems 1–5 to find the particular equation of the cubic function if the leading coefficient is 1 and the zeros are 7. Find the particular equation of the cubic function that is a vertical dilation of by a factor of 3. 8. Plot the graphs of g and h on the same screen. Do both functions have the same zeros? Sketch the graphs here. 9. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-3a: Fitting a Polynomial Date:
Function to Points

Objective: Given a set of points, find the particular equation of a polynomial
function that fits the points.

 The figure shows the graph of a polynomial function P. The table shows plotting points for this function. 1. Show that the third differences between the values are constant. You may do this by entering the x- and y-data in lists on your grapher and using operations on the y -list to make lists of first, second, and third differences. Third differences = 2. If the third differences between y-values are constant, then a cubic function fits the data. Find the particular equation of the cubic function that fits the first four data points. You may do this by substituting 1, 2, 3, and 4 for x and then solving for a, b, c, and d by matrices. 3. Show that the equation of Problem 2 fits the other four data points. 4. Find the particular equation of Problem 2 again, using the cubic regression feature of your grapher. Does the equation come out the same? What statistic tells you that the fit is perfect ?5. Use your equation to predict the value of P(20). 6. Find numerically the largest value of x for which P(x) = 10. Does the answer agree with the graph? 7. The number 4 is said to be a zero of P(x)because P(4) = 0.  This fact implies that (x - 4) is a factor of P(x)P(x) = (x - 4)(other factor) Find the other (quadratic) factor. 8. By setting the quadratic factor in Problem 7 equal to zero, find algebraically the other two zeros of P(x). Do the answers agree with the graph? 9. What did you learn as a result of doing this Exploration that you did not know before?

Exploration 15-4a: Rational Functions Date:
and Discontinuities

Objective: Find and classify discontinuities in the graph of a rational algebraic function.

 1. On this graph paper, plot quickly the graph of the rational function (no grapher). 2. The graph in Problem 1 has a vertical asymptote at x = 0. Give an algebraic reason why there is such an asymptote there. 3. Identify a transformation that maps the graph of f onto the graph of g shown here. (There are at least three ways to answer this!) 4. Let What transformation of the graph of f is this? On the graph paper here, plot quickly the graph of h (no grapher). 5. Function r is called a rational function because equals a ratio of two polynomials. Plot the graph of r on your grapher. Use a friendly window from about to that includes each integer as a grid point. Sketch the result on this graph paper. 6. In what way is the graph of r similar to the graph of h in Problem 4? In what way is it different ? 7. The graph of r has a removable discontinuity at . Explain algebraically why there is a discontinuity here. 8. By factoring the denominator in the equation for , show how the discontinuity at can be “removed” algebraically. 9. Without plotting the graph, how can you tell which kind of discontinuity, removable or asymptote, the graph of a rational function will have at a value of x that makes the denominator equal zero? 10. What did you learn as a result of doing this Exploration that you did not know before?
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