Julie has tracked Bouncy Ball company profits for several years, and her analysis has determined that the function, P(x) = –3x3 + 138x2 + 588x + 600, can be used to determine profits, in dollars, every year for x years after 1980. 1) Find the y–intercept for the profit function and give an interpretation. 2) Determine the end behavior of the profit function. a) What is the behavior at the left end? Is this behavior relevant to company profits? If so, what does it say about company profits as x → – ∞? b) What is the behavior at the right end? Is this behavior relevant to company profits? If so, what does it say about company profits as x → ∞?
Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Julie has tracked Bouncy Ball company profits for several
years, and her analysis has determined that the function,
P(x) = –3x3 + 138x2 + 588x + 600, can be used to determine
profits, in dollars, every year for x years after 1980.
1) Find the y–intercept for the profit function and give an interpretation.
2) Determine the end behavior of the profit function.
a) What is the behavior at the left end? Is this behavior relevant to
company profits? If so, what does it say about company profits as
x → – ∞?
b) What is the behavior at the right end? Is this behavior relevant to
company profits? If so, what does it say about company profits as
x → ∞?
3) We will use the Rational Zeros Theorem to determine possible rational zeros
of P(x). First, factor out the common factor of -3.
4) Use synthetic division to find at least one rational zero of P(x).
Chapter 5: Polynomial Functions 69
Profitable Venture Page 2 of 2
5) Use the zero found in item 4) along with the quotient to completely factor
P(x).
6) What are the x–intercepts of P(x)?
7) Use the zeros, y–intercept, and end–behavior of P(x) to draw a sketch of the
function. Use your calculator to verify that your sketch is correct.
8) Use the graph to determine an appropriate domain and range for this function.
When does the profit model become ineffective?
a) What is an appropriate domain?
b) What is an appropriate range?
c) When is the model ineffective? Why?
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