Exam 3: Polynomial and Rational Functions

Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible. -Crosses the $x$ -axis at $- 1,0$ , and 3 ; lies above the $x$ -axis between $- 1$ and 0 ; lies below the $x$ -axis between 0 and 3 .

(Multiple Choice)

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Question 101

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = ( x + 5 ) ^ { 2 } + 5$

(Multiple Choice)

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Question 102

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = x ^ { 2 } + 16 x + 41$ Give your answers in exact form.

(Multiple Choice)

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Question 103

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = 6 + 5 x + x ^ { 2 }$

(Multiple Choice)

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Question 104

Find the zeros of the polynomial function - $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - x - 5$

(Multiple Choice)

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Question 105

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. -

(Multiple Choice)

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Question 106

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = x ^ { 2 } - 5$

(Multiple Choice)

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Question 107

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers - $f ( x ) = 6 x ^ { 4 } - 3 x ^ { 3 } + 4 x - 3$ ; between $- 1$ and 0

(Multiple Choice)

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Question 108

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. Determine the degree of the polynomial function of best fit and the sign of the leading coefficient.

(Multiple Choice)

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Question 109

The graph of a quadratic function is given. Determine the function's equation. -

(Multiple Choice)

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Question 110

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 3$

(Multiple Choice)

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Question 111

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$

(Multiple Choice)

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Question 112

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = 4 x ^ { 3 } + 5 x ^ { 3 } - x ^ { 5 }$

(Multiple Choice)

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Question 113

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. - $f ( x ) = x ^ { 4 } - 4 x ^ { 2 }$

(Multiple Choice)

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Question 114

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function decreasing?

(Multiple Choice)

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Question 115

Find the x-intercepts (if any) for the graph of the quadratic function. - $5 x ^ { 2 } + 12 x + 3 = 0$ Give your answers in exact form.

(Multiple Choice)

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Question 116

Find the axis of symmetry of the parabola defined by the given quadratic function. - $f ( x ) = ( x + 4 ) ^ { 2 } + 7$

(Multiple Choice)

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Question 117

Determine whether the function is a polynomial function. - $( x ) = \frac { 2 - x ^ { 5 } } { 3 }$

(Multiple Choice)

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Question 118

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. - $f ( x ) = 5 \left( x ^ { 2 } + 2 \right) ( x + 1 ) ^ { 2 }$

(Multiple Choice)

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Question 119

Solve the problem. -Write an equation in standard form of the parabola that has the same shape as the graph of f(x)= -7x² , but which has a maximum of 5 at x=3 .

(Multiple Choice)

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Question 120

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = ( x + 5 ) ^ { 2 } + 5$

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = x ^ { 2 } + 16 x + 41$ Give your answers in exact form.

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = 6 + 5 x + x ^ { 2 }$

Find the zeros of the polynomial function - $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - x - 5$

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. -

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = x ^ { 2 } - 5$

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers - $f ( x ) = 6 x ^ { 4 } - 3 x ^ { 3 } + 4 x - 3$ ; between $- 1$ and 0

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. Determine the degree of the polynomial function of best fit and the sign of the leading coefficient.

The graph of a quadratic function is given. Determine the function's equation. -

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 3$

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = 4 x ^ { 3 } + 5 x ^ { 3 } - x ^ { 5 }$

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. - $f ( x ) = x ^ { 4 } - 4 x ^ { 2 }$

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function decreasing?

Find the x-intercepts (if any) for the graph of the quadratic function. - $5 x ^ { 2 } + 12 x + 3 = 0$ Give your answers in exact form.

Find the axis of symmetry of the parabola defined by the given quadratic function. - $f ( x ) = ( x + 4 ) ^ { 2 } + 7$

Determine whether the function is a polynomial function. - $( x ) = \frac { 2 - x ^ { 5 } } { 3 }$

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. - $f ( x ) = 5 \left( x ^ { 2 } + 2 \right) ( x + 1 ) ^ { 2 }$

Solve the problem. -Write an equation in standard form of the parabola that has the same shape as the graph of f(x)= -7x² , but which has a maximum of 5 at x=3 .

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Exam 3: Polynomial and Rational Functions

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - f(x)=(x+5)2+5f ( x ) = ( x + 5 ) ^ { 2 } + 5f(x)=(x+5)2+5

Find the x-intercepts (if any) for the graph of the quadratic function. - f(x)=x2+16x+41f ( x ) = x ^ { 2 } + 16 x + 41f(x)=x2+16x+41 Give your answers in exact form.

Find the x-intercepts (if any) for the graph of the quadratic function. - f(x)=6+5x+x2f ( x ) = 6 + 5 x + x ^ { 2 }f(x)=6+5x+x2

Find the zeros of the polynomial function - f(x)=x3+5x2−x−5f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - x - 5f(x)=x3+5x2−x−5

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. -

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - f(x)=x2−5f ( x ) = x ^ { 2 } - 5f(x)=x2−5

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers - f(x)=6x4−3x3+4x−3f ( x ) = 6 x ^ { 4 } - 3 x ^ { 3 } + 4 x - 3f(x)=6x4−3x3+4x−3 ; between −1- 1−1 and 0

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. Determine the degree of the polynomial function of best fit and the sign of the leading coefficient.

The graph of a quadratic function is given. Determine the function's equation. -

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - f(x)=−3x3−2x2+3x+3f ( x ) = - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 3f(x)=−3x3−2x2+3x+3

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. - f(x)=−x2(x+6)(x2−1)f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)f(x)=−x2(x+6)(x2−1)

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - f(x)=4x3+5x3−x5f ( x ) = 4 x ^ { 3 } + 5 x ^ { 3 } - x ^ { 5 }f(x)=4x3+5x3−x5

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. - f(x)=x4−4x2f ( x ) = x ^ { 4 } - 4 x ^ { 2 }f(x)=x4−4x2

Solve. -Suppose that a polynomial function is used to model the data shown in the graph below. For what intervals is the function decreasing?

Find the x-intercepts (if any) for the graph of the quadratic function. - 5x2+12x+3=05 x ^ { 2 } + 12 x + 3 = 05x2+12x+3=0 Give your answers in exact form.

Find the axis of symmetry of the parabola defined by the given quadratic function. - f(x)=(x+4)2+7f ( x ) = ( x + 4 ) ^ { 2 } + 7f(x)=(x+4)2+7

Determine whether the function is a polynomial function. - (x)=2−x53( x ) = \frac { 2 - x ^ { 5 } } { 3 }(x)=32−x5​

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. - f(x)=5(x2+2)(x+1)2f ( x ) = 5 \left( x ^ { 2 } + 2 \right) ( x + 1 ) ^ { 2 }f(x)=5(x2+2)(x+1)2

Solve the problem. -Write an equation in standard form of the parabola that has the same shape as the graph of f(x)= -7x2 , but which has a maximum of 5 at x=3 .

Equations and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

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Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = ( x + 5 ) ^ { 2 } + 5$

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = x ^ { 2 } + 16 x + 41$ Give your answers in exact form.

Find the x-intercepts (if any) for the graph of the quadratic function. - $f ( x ) = 6 + 5 x + x ^ { 2 }$

Find the zeros of the polynomial function - $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - x - 5$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = x ^ { 2 } - 5$

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers - $f ( x ) = 6 x ^ { 4 } - 3 x ^ { 3 } + 4 x - 3$ ; between $- 1$ and 0

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 3$

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } - 1 \right)$

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. - $f ( x ) = 4 x ^ { 3 } + 5 x ^ { 3 } - x ^ { 5 }$

Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither. - $f ( x ) = x ^ { 4 } - 4 x ^ { 2 }$

Find the x-intercepts (if any) for the graph of the quadratic function. - $5 x ^ { 2 } + 12 x + 3 = 0$ Give your answers in exact form.

Find the axis of symmetry of the parabola defined by the given quadratic function. - $f ( x ) = ( x + 4 ) ^ { 2 } + 7$

Determine whether the function is a polynomial function. - $( x ) = \frac { 2 - x ^ { 5 } } { 3 }$

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. - $f ( x ) = 5 \left( x ^ { 2 } + 2 \right) ( x + 1 ) ^ { 2 }$

Solve the problem. -Write an equation in standard form of the parabola that has the same shape as the graph of f(x)= -7x² , but which has a maximum of 5 at x=3 .