AP®MathsCollege BoardPrecalculusExam QuestionsPolynomial & Rational FunctionsLinear, Quadratic & Polynomial Functions

Linear, Quadratic & Polynomial Functions (College Board AP® Precalculus): Exam Questions

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Linear, Quadratic & Polynomial Functions

1
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2 marks

The function g is given by g open parentheses x close parentheses equals negative 0.167 x cubed plus x squared minus 1.834.

(i) Find all values of x, as decimal approximations, for which g open parentheses x close parentheses equals 0, or indicate that there are no such values.

(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.

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2
1 mark

A bicycle rental company charges a flat fee plus a constant rate per hour for bicycle rentals. The cost of a 2-hour rental is $14, and the cost of a 5-hour rental is $26.

The total cost, in dollars, of a bicycle rental can be modeled by the function C given by C open parentheses t close parentheses equals a plus b t, where C open parentheses t close parentheses is the total cost, in dollars, and t is the number of hours of the rental.

Use the given data to write two equations that can be used to find the values for constants a and b in the expression for C open parentheses t close parentheses.

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3
1 mark

The table gives values for a function space f at selected values of x.

space x

0

1

2

3

4

space f open parentheses x close parentheses

5

3

5

11

21

Based on the table, which of the following function types best models function space f: linear, quadratic, exponential, or logarithmic?

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4
1 mark

The polynomial function space f is given by space f open parentheses x close parentheses equals negative 2 x to the power of 4 plus 5 x squared minus 1.

Determine the end behavior of space f as x increases without bound. Express your answer using the mathematical notation of a limit.

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5
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2 marks

The function g is given by g open parentheses x close parentheses equals negative 0.1371 x to the power of 4 plus x cubed minus 1.2 x squared.

(i) Find all values of x, as decimal approximations, for which g open parentheses x close parentheses equals 11, or indicate that there are no such values.

(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.

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6a
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1 mark

The polynomial function space p is given by space p open parentheses x close parentheses equals x cubed minus 4 x squared plus 4 x. Rewrite space p open parentheses x close parentheses as a product of linear factors.

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6b
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2 marks

Find all real zeros of space p, and state the multiplicity of each. For each zero, state whether the graph of y equals p open parentheses x close parentheses crosses the x-axis or is tangent to the x-axis at that point.

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6c
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1 mark

Determine the end behavior of space p. Express your answer using the mathematical notation of a limit.

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7
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2 marks

Use Pascal's Triangle to rewrite \left(x - 2\right)^{4} in the form a x^{4} + b x^{3} + c x^{2} + d x + e, where a, b, c, d, and e are integers to be found.

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8
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2 marks

The polynomial function space p is odd, and space p has a relative minimum value of negative 7 at x equals 3. Find the value of space p open parentheses negative 3 close parentheses, and state whether space p open parentheses negative 3 close parentheses is a relative maximum or a relative minimum of space p. Give a reason for your answer.

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9a
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1 mark

The height of a ball, in meters, t seconds after it is dropped from a building is given in the table.

t (seconds)

0

1

2

3

4

h open parentheses t close parentheses (meters)

100

95.1

80.4

55.9

21.6

Find the average rate of change of h over the interval open square brackets 0 comma 1 close square brackets, and the average rate of change of h over the interval open square brackets 3 comma 4 close square brackets. Show the work that leads to your answers.

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9b
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1 mark

Based on the table, determine which type of function best models h: linear, quadratic, cubic, or quartic. Give a reason for your answer based on the relationship between the change in the output values of h and the change in the input values of h.

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9c
1 mark

Compare the average rates of change you found in part (a). Interpret the meaning of this comparison in the context of the problem.

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10a
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1 mark

The polynomial function space p is given by space p open parentheses x close parentheses equals x cubed minus x squared minus 6 x. Rewrite space p open parentheses x close parentheses as a product of linear factors.

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10b
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1 mark

Find all values of x for which space p open parentheses x close parentheses greater than 0.

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11a
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1 mark

Maria opens a savings account with an initial deposit, and then makes equal weekly deposits of the same amount each week. After 2 weeks, the account has $250 in it. After 8 weeks, the account has $670 in it.

Let A open parentheses t close parentheses be the amount of money in Maria's savings account, in dollars, t weeks after she opened it. Construct a linear model for A open parentheses t close parentheses.

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11b
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1 mark

Interpret the meaning of the slope of your linear model in the context of this problem. Include appropriate units in your interpretation.

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12a
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2 marks

The water level in a reservoir, in meters, is modeled by the function L, where L open parentheses t close parentheses equals negative t cubed plus 11 t squared plus 100 for 0 less or equal than t less or equal than 10. L open parentheses t close parentheses is the water level in meters, and t is the number of days since the start of the rainy season.

Find the maximum water level during this period, and the time at which the maximum occurs. Express both values as decimal approximations.

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12b
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1 mark

Find the average rate of change of L over the interval open square brackets 0 comma 5 close square brackets. Include appropriate units in your answer. Show the work that leads to your answer.

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12c
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1 mark

Interpret the meaning of your answer to part (b) in the context of the problem.

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13
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2 marks

A polynomial function space p has degree 3 with real coefficients. Two of the zeros of space p are x equals 4 and x equals negative 1 plus 2 i. The graph of space p passes through the point open parentheses 0 comma negative 20 close parentheses. Construct an expression for space p open parentheses x close parentheses in standard form. Show the work that leads to your answer.

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14
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3 marks

The polynomial function space f is given by space f open parentheses x close parentheses equals x to the power of 4 minus 6 x cubed plus 7 x squared plus 12 x minus 18.

(i) Find all values of x, as decimal approximations, for which space f open parentheses x close parentheses equals 0, or indicate that there are no such values.

(ii) Determine the end behavior of space f as x decreases without bound. Express your answer using the mathematical notation of a limit.

(iii) For each zero of space f found in (i), determine whether the multiplicity of the zero is even or odd. Give a reason for your answer.

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