Mathematics Practices
Exploration 151a: Cubic Function Graphs
Objective: Learn the shape and some properties of cubic function graphs.
1. Let Plot the graph using a window with xrange of about [5, 5] and a yrange of [30, 30]. Sketch the graph.
2. The graph in Problem 1 crosses the xaxis at
three 3. The integer value in Problem 2 can be used to
write a 4. With the help of the quadratic formula , find
the 5. Let Plot
the graph of g on 
6. Factor into three linear factors .
By setting each factor equal to zero, find the three xintercepts. What do you notice about two of the xintercepts?
7. Let Plot the
graph of h. 8. Find one linear factor of
. Use it to find two other 9. What did you learn as a result of doing this

Exploration 152a: 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 xrange of about [5, 5] and a yrange of [100, 100]. Sketch the graph here.
2. Find . Mark
the corresponding point on your 3. Use long division to divide
by
See 4. What do you notice about the remainder and the 
6. How could you write the answer to Problem 3 directly from the synthetic substitution results ?
7. Find by
synthetic substitution. How does the 8. Use the result of Problem 7 to find the other
two 9. What did you learn as a result of doing this

Exploration 152b: 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. 2. Find the product of the zeros, 3. If you factor out 5 from each term in the
equation of How does the product of the zeros you found in 4. Find the sum of the zeros,
.How does the 
5. Find the sum of the pairwise products of the
zeros,
How does the answer relate to the coefficients in 6. Use the patterns you observe in Problems 1–5 to
find 7. Find the particular equation of the cubic
function 8. Plot the graphs of g and h on the same screen.
Do 9. What did you learn as a result of doing this

Exploration 153a: 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
Find the particular equation of the cubic function 3. Show that the equation of Problem 2 fits the
other 
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). P(x) = (x  4)(other factor)

Exploration 154a: 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).

5. Function r is called a rational function because
equals a ratio of two polynomials.
10. What did you learn as a result of doing this

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