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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1 mark

The polynomial function space p is given by space p open parentheses x close parentheses equals 4 minus 3 x squared plus 7 x minus 2 x to the power of 4. What are the degree and leading coefficient of space p?

  • The degree is 2 and the leading coefficient is - 3.

  • The degree is 4 and the leading coefficient is - 3.

  • The degree is 4 and the leading coefficient is - 2.

  • The degree is 4 and the leading coefficient is 4.

Choose your answer

2
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1 mark

The table gives values for four functions, space f, g, h, and k, at selected values of x.

space x

0

2

4

6

space f open parentheses x close parentheses

1

4

9

16

space g open parentheses x close parentheses

1

5

9

13

space h open parentheses x close parentheses

2

6

18

54

space k open parentheses x close parentheses

3

7

13

21

Which of the following could be a linear function?

  • space f

  • g

  • h

  • k

Choose your answer

3
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1 mark

The polynomial function space p is given in factored form by space p open parentheses x close parentheses equals open parentheses x minus 1 close parentheses open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses.

Which of the following is equivalent to space p open parentheses x close parentheses?

  • x^{3} - 2 x^{2} - 5 x - 6

  • x^{3} - 2 x^{2} - 5 x + 6

  • x^{3} - 2 x^{2} + 5 x + 6

  • x^{3} + 2 x^{2} - 5 x + 6

Choose your answer

4
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1 mark

What is the coefficient of x cubed in the expansion of open parentheses x minus 2 close parentheses to the power of 5?

  • - 80

  • 4

  • 20

  • 40

Choose your answer

5
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1 mark

A polynomial function q is given by q open parentheses x close parentheses equals x open parentheses x minus 5 close parentheses open parentheses x plus 3 close parentheses. What are all intervals on which q open parentheses x close parentheses less or equal than 0 ?

  • \left[0 , 5\right]

  • \left[- 3 , 5\right]

  • \left(- \infty , - 3\right] \cup \left[0 , 5\right]

  • \left[- 3 , 0\right] \cup \left[5 , \infty\right)

Choose your answer

6
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1 mark

The function g is given by g \left(x\right) = - 3 x^{5} + 4 x^{2} - 1. Which of the following describes the end behavior of g ?

  • \underset{x \rightarrow - \infty}{\lim} g \left(x\right) = - \infty and \underset{x \rightarrow \infty}{\lim} g \left(x\right) = - \infty

  • \underset{x \rightarrow - \infty}{\lim} g \left(x\right) = \infty and \underset{x \rightarrow \infty}{\lim} g \left(x\right) = \infty

  • \underset{x \rightarrow - \infty}{\lim} g \left(x\right) = - \infty and \underset{x \rightarrow \infty}{\lim} g \left(x\right) = \infty

  • \underset{x \rightarrow - \infty}{\lim} g \left(x\right) = \infty and \underset{x \rightarrow \infty}{\lim} g \left(x\right) = - \infty

Choose your answer

7
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1 mark

The polynomial function p is given by p \left(x\right) = \left(x - 2\right) \left(x^{2} + x + 7\right). Which of the following describes the zeros of p ?

  • p has exactly three distinct real zeros.

  • p has exactly two distinct real zeros.

  • p has exactly one distinct real zero and no non-real zeros.

  • p has exactly one distinct real zero and two non-real zeros.

Choose your answer

8
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1 mark

The table shows values for a function h at selected values of x.

x

0

1

2

3

4

h \left(x\right)

12

40

62

78

88

Which of the following claim and explanation statements best fits these data?

  • h is best modeled by a linear function, because the rate of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a linear function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a quadratic function, because the rate of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a quadratic function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

Choose your answer

9
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1 mark

The polynomial function k is given by k \left(x\right) = a x^{4} - b x^{3} + 21, where a and b are nonzero real constants. Each of the zeros of k has multiplicity 1. In the x y-plane, an x-intercept of the graph of k is open parentheses 12.458 comma space 0 close parentheses. A zero of k is - 1.23 - 0.56 i. Which of the following statements must be true?

  • The graph of k has three x-intercepts.

  • - 1.23 + 0.56 i is a zero of k.

  • The equation k \left(x\right) = 0 has four real solutions.

  • The graph of k is tangent to the x-axis at x = 12.458.

Choose your answer

10
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1 mark

The polynomial function p is given by p \left(x\right) = 3x^{6} - 4x + 2. Which of the following statements about the end behavior of p is true?

  • The sign of the leading term of p is positive, and the degree of the leading term is even; therefore limit as x rightwards arrow negative infinity of p left parenthesis x right parenthesis equals infinity and limit as x rightwards arrow infinity of p left parenthesis x right parenthesis equals infinity.

  • The sign of the leading term of p is positive, and the degree of the leading term is odd; therefore limit as x rightwards arrow negative infinity of p left parenthesis x right parenthesis equals negative infinity and limit as x rightwards arrow infinity of p left parenthesis x right parenthesis equals infinity.

  • The sign of the leading term of p is negative, and the degree of the leading term is even; therefore limit as x rightwards arrow negative infinity of p left parenthesis x right parenthesis equals negative infinity and limit as x rightwards arrow infinity of p left parenthesis x right parenthesis equals negative infinity.

  • The sign of the leading term of p is negative, and the degree of the leading term is odd; therefore limit as x rightwards arrow negative infinity of p left parenthesis x right parenthesis equals infinity and limit as x rightwards arrow infinity of p left parenthesis x right parenthesis equals negative infinity.

Choose your answer

11
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1 mark

The polynomial function p is an even function. If p(5) = -3 is a relative minimum of p, which of the following statements about p(-5) must be true?

  • p(-5) = 3 and p(-5) is a relative maximum

  • p(-5) = -3 and p(-5) is a relative maximum

  • p(-5) = 3 and p(-5) is a relative minimum

  • p(-5) = -3 and p(-5) is a relative minimum

Choose your answer

12
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1 mark
A graph labelled "Graph of f" on the xy-plane with arrowheads on both axes. The curve of f enters from the upper left, crosses the x-axis at x = −4 going from positive to negative, reaches a minimum of approximately −128 near x = −2.5, rises back to touch the x-axis tangentially at x = 2 (a local maximum touching from below without crossing), dips to a second minimum of approximately −32 near x = 4, then crosses the x-axis at x = 6 going from negative to positive, and rises steeply toward the upper right. Both ends of the graph rise toward +∞. Tick marks are shown on the x-axis at −4, −2, 0, 2, 4, 6 and on the y-axis at −100, −50, 50.

The figure shows the graph of a polynomial function f. The graph crosses the x-axis at x = -4 and x = 6, touches but does not cross the x-axis at x = 2, and both ends of the graph rise toward +\infty. Which of the following could be an expression for f(x)?

  • 0.5(x + 4)(x - 2)(x - 6)

  • 0.5(x + 4)^2(x - 2)(x - 6)

  • 0.5(x + 4)(x - 2)^2(x - 6)

  • 0.5(x + 4)(x - 2)^2(x + 6)

Choose your answer

13
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1 mark

The number of customers arriving at a coffee shop on a particular day is modeled by the function S, where S open parentheses t close parentheses equals 0.04 t cubed minus 0.96 t squared plus 6.3 t plus 1.8 for 0 less or equal than t less or equal than 16. S \left(t\right) is measured in hundreds of customers per hour, and t is measured in hours since the shop opened. Based on the model, at what value of t does the rate of customers arriving change from increasing to decreasing?

  • t = 1 . 800

  • t = 4 . 609

  • t = 11 . 391

  • t = 16

Choose your answer

14
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1 mark

The table shows values for a function h at selected values of x.

x

0

1

2

3

4

h\left(x\right)

3

10

21

36

55

Which of the following claim and explanation statements best fits these data?

  • h is best modeled by a linear function, because the rate of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a linear function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a quadratic function, because the rate of change over consecutive equal-length input-value intervals is constant.

  • h is best modeled by a quadratic function, because the change in the average rates of change over consecutive equal-length input-value intervals is constant.

Choose your answer

15
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1 mark

The number of visitors entering a theme park on a particular day is modeled by the function V, where V \left(t\right) = 0.05 t^{3} - 1.2 t^{2} + 8.1 t + 2.5 for 0 \leq t \leq 16. V \left(t\right) is measured in hundreds of visitors per hour, and t is measured in hours since the park opened. Based on the model, at what value of t does the rate of visitors entering the park change from increasing to decreasing?

  • t = 16

  • t = 11.162

  • t = 4.838

  • t = 2.500

Choose your answer

16
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1 mark

The table shows values for a function g at selected values of x.

x

0

1

2

3

4

5

g \left(x\right)

1

0

1

10

33

76

Based on the table, which type of function best models g?

  • Linear

  • Quadratic

  • Cubic

  • Quartic

Choose your answer

17
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1 mark

The polynomial function space p is given by space p open parentheses x close parentheses equals open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses open parentheses x squared plus 1 close parentheses. Which of the following statements about space p is true?

  • space p is even, because space p open parentheses negative x close parentheses equals p open parentheses x close parentheses for all real values of x.

  • space p is neither even nor odd, because the graph of space p does not pass through the origin.

  • space p is neither even nor odd, because space p open parentheses negative x close parentheses equals open parentheses negative x minus 2 close parentheses open parentheses negative x plus 2 close parentheses open parentheses negative x squared plus 1 close parentheses, which is not equal to space p open parentheses x close parentheses.

  • space p is odd, because space p open parentheses negative x close parentheses equals negative p open parentheses x close parentheses for all real values of x.

Choose your answer

18
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1 mark

The polynomial function space f is defined for all real numbers by space f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses open parentheses x minus 1 close parentheses squared.

What are all intervals on which space f open parentheses x close parentheses greater than 0?

  • \left(- \infty , - 2\right) \cup \left(1 , \infty\right)

  • \left(- 2 , 1\right)

  • \left(- 2 , 1\right) \cup \left(1 , \infty\right)

  • \left(- 2 , \infty\right)

Choose your answer

19
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1 mark

The polynomial function space f is defined by space f open parentheses x close parentheses equals negative 2 open parentheses x minus 3 close parentheses squared open parentheses x plus 1 close parentheses cubed.

Which of the following statements about the end behavior of space f is true?

  • The sign of the leading term of space f is negative, and the degree of the leading term of space f is even; therefore, limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals negative infinity and limit as x rightwards arrow infinity of f open parentheses x close parentheses equals negative infinity.

  • The sign of the leading term of space f is negative, and the degree of the leading term of space f is odd; therefore, limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals infinity and limit as x rightwards arrow infinity of f open parentheses x close parentheses equals negative infinity.

  • The sign of the leading term of space f is positive, and the degree of the leading term of space f is even; therefore, limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals infinity and limit as x rightwards arrow infinity of f open parentheses x close parentheses equals infinity.

  • The sign of the leading term of space f is positive, and the degree of the leading term of space f is odd; therefore, limit as x rightwards arrow negative infinity of f open parentheses x close parentheses equals negative infinity and limit as x rightwards arrow infinity of f open parentheses x close parentheses equals infinity.

Choose your answer

20
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1 mark

The polynomial function space f is defined by space f open parentheses x close parentheses equals open parentheses x minus a close parentheses squared open parentheses x minus b close parentheses, where a and b are real numbers with a less than b. At which value of x does space f have a local maximum?

  • x = a

  • x = \dfrac{a + b}{2}

  • x = b

  • The location cannot be determined.

Choose your answer

21
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1 mark

In the expansion of \left(2 x - 3 y\right)^{5}, what is the coefficient of x^{2} y^{3}?

  • -1080

  • -720

  • -540

  • 1080

Choose your answer

22
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1 mark

The function space f is defined by space f open parentheses x close parentheses equals x cubed minus 2 x squared minus 5 x plus 6, and x equals 1 is a zero of space f. Let a and b denote the other two real zeros of space f. What is the value of open vertical bar a minus b close vertical bar?

  • 1

  • 4

  • 5

  • 7

Choose your answer

23
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1 mark

The graph of the quadratic function space f open parentheses x close parentheses equals a x squared plus b x plus c has vertex at open parentheses 2 comma negative 3 close parentheses and passes through the point open parentheses 0 comma 5 close parentheses. What is the value of a plus b plus c?

  • -3

  • -1

  • 5

  • 7

Choose your answer