Exam 3: Polynomial and Rational Functions

Solve the problem. -The cost in millions of dollars for a company to manufacture $x$ thousand automobiles is given by the function $C ( x ) = 3 x ^ { 2 } - 30 x + 200$ . Find the number of automobiles that must be produced to minimize the cost.

Divide using synthetic division. - $\left( x ^ { 2 } + 13 x + 38 \right) \div ( x + 7 )$

Find the vertical asymptotes, if any, of the graph of the rational function. - $f ( x ) = \frac { x + 5 } { x ( x - 5 ) }$

Use synthetic division and the Remainder Theorem to find the indicated function value. - $f ( x ) = 5 x ^ { 4 } + 7 x ^ { 3 } + 6 x ^ { 2 } - 4 x + 51 ; f ( 4 )$

Determine the constant of variation for the stated condition. -For a resistor in a direct current circuit that does not vary its resistance, the power that a resistor must dissipate is directly proportional to the square of the voltage across the resistor. The resistor must dissipate $\frac { 1 } { 16 }$ watt of power when the voltage across the resistor is 7 volts. Find the power that the resistor must dissipate when the voltage across it is 14 volts.

Find the indicated intercept(s) of the graph of the function. -Is there $y$ -axis symmetry for the rational function $f ( x ) = \frac { - 6 x ^ { 2 } } { 2 x ^ { 4 } - 12 }$ ?

Use synthetic division and the Remainder Theorem to find the indicated function value. - $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } - 5 x ^ { 2 } + 8 x - 4 ; f \left( \frac { 1 } { 2 } \right)$

Find the y-intercept of the polynomial function. - $f ( x ) = x ^ { 3 } - 5 x$

Solve the problem. -The average cost per unit, $\mathrm { y }$ , of producing $x$ units of a product is modeled by $y = \frac { 450,000 + 0.55 x } { x }$ . Describe the company's production level so that the average cost of producing each unit does not exceed $\$ 5.05$ .

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. - $f ( x ) = 2 x ^ { 2 } - 4 x$

Find the axis of symmetry of the parabola defined by the given quadratic function. - $f ( x ) = - 7 ( x - 2 ) ^ { 2 } - 9$

Determine the constant of variation for the stated condition. -Body-mass index, or BMI, takes both weight and height into account when assessing whether an individual is underweight or overweight. BMI varies directly as one's weight, in pounds, and inversely as the square of one's height, in inches. In adults, normal values for the BMI are between 20 and 25. A person who weighs 172 pounds and is 69 inches tall has a BMI of 25.4. What is the BMI, to the nearest tenth, for a person who weighs 120 pounds and who is 65 inches tall?

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end - $f ( x ) = 3 x ^ { 3 } - 3 x ^ { 2 } - 5 x - 2$

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. - $f ( x ) = - x ^ { 2 } ( x + 7 ) \left( x ^ { 2 } + 1 \right)$

Determine the constant of variation for the stated condition. -If $y$ varies directly as $x$ , and $y = 3$ when $x = 5$ , find $y$ when $x = 20$ .

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. - $f ( x ) = - 3 x ^ { 2 } + 9 x$

Write an equation that expresses the relationship. Use k as the constant of variation. -The amount of time it takes a swimmer to swim a race is inversely proportional to the average speed of the swimmer. A swimmer finishes a race in 200 seconds with an average speed of 3 feet per second. Find the average speed of the swimmer if it takes 120 seconds to finish the race.

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $f ( x ) = ( x + 2 ) ^ { 2 } + 8$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. - $y + 9 = ( x - 3 ) ^ { 2 }$

Find the y-intercept for the graph of the quadratic function. - $y + 9 = ( x + 3 ) ^ { 2 }$