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Math Topics

Precalculus

Polynomial and Rational Functions

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1
Quadratic Functions and Their Graphs

Let \(f(x)=-3x^2-6x+2\)

Fins the maximum value of \(f\).

Find the axis os symmetry of the parabola.

Find the \(x\)-intercepts, if they exist.

Find two additional points on the graph, and then sketch the graph of \(f\) by hand.

Use the graph to find intervals where \(f\) is increasing or decreasing and find the range.

Traffic authorities have \(100\) feet of rope to cordon off a rectangular region to form a ticket arena specifically for concert goers who are waiting to purchase tickets.

Express the area of this region as a function of the length of just one of the four sides of the region.

Find the dimensions of the enclosed region that will give the maximum area, and determine the maximum area.

The following table lists the speed at which a car is driven, in miles per hour, and the corresponding gas mileage obtained, in miles per gallon.

Make a scatter plot of \(m\), the gas mileage, vs. \(x\), the speed at which a car is driven. What type of trend do you observe - linear or quadratic? Find an expression for the function that best fits the given data points.

Use your function to find the gas mileage obtained when a car is driven at \(35\) miles per hour.

Find the speed at which the carâ€™s gas mileage is at its maximum, using a graphing utility.

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2
Polynomial Functions and Their Graphs

Gift Horse, Inc., manufactures various type of decorative gift boxes. The bottom portion of one such box is made by cutting a small square of length \(x\) inches from each corner of a \(10\)-inch piece of cardboard and folding up the sides of the box. See Figure 16.

Find an expression for the volume of the resulting box.

Graph the volume function. Use your graph to determine the values of \(x\) for which the expression makes realistic sense.

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3
Division of Polynomials; the Remainder and Factor Theorems

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4
Real Zeros of Polynomials; Solutions of Equations

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Complex Numbers

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6
The Fundamental Theorem of Algebra; Complex Zeros

Find a polynomial \(p(x)\) of degree \(4\) with \(p(0)=-9\) and zeros \(x=-3\), \(x=1\), and \(x=3\), with \(x=3\) a zero of multiplicity \(2\). For this polynomial, is it possible for the zeros other than \(3\) to have a multiplicity greater than \(1\)?

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7
Rational Functions

Find the horizontal asymptote, if it exists, for each of the following rational functions. Use a table and a graph to discuss the end behavior of each function.

\(f(x)=\dfrac{x+2}{x^2-1}\)

\(f(x)=\dfrac{2x}{x+1}\)

\(f(x)=\dfrac{x^2-3}{2x+1}\)

Suppose it costs \(\$45\) a day to rent a car with unlimited mileage.

What is the expression for the average cost per mile per day?

Make a table and graph of the average cost function.

What happens to the average cost per day as the number of miles driven per day increases?