Understanding Coefficients and Transformations in Cubic Functions

A cubic function is modeled by the equation f(x) = x^3 - 3x^2 + 2. What do the coefficients of this function tell you about its shape and behavior?

• The function is linear with no local extrema.

• The function has two local maxima and no local minima.

• The function has a cubic shape with one local maximum and one local minimum, rising to positive infinity and falling to negative infinity.

• The function is a quadratic with a parabolic shape.

The volume of a cube is given by the function V(x) = x^3. If the side length of the cube is increased by 2 units, what is the new volume?

• (x + 2)^3 = x^3 + 12x + 4

• (x + 2)^3 = x^3 + 4x + 4

• (x + 2)^3 = x^3 + 8x + 16

• (x + 2)^3 = x^3 + 6x^2 + 12x + 8

A company's profit can be modeled by the cubic function P(x) = -2x^3 + 12x^2 + 3. What does the coefficient of x^3 indicate about the profit as production increases?

• The positive coefficient indicates that profit increases as production increases.

• The coefficient of x^3 suggests profit remains constant regardless of production levels.

• The negative coefficient indicates that profit decreases as production increases.

• The coefficient of x^3 has no effect on profit as production increases.

The function f(x) = 2(x - 1)^3 + 4 represents the height of a water tank over time. How does the transformation affect the height compared to the basic cubic function?

• The height remains unchanged and is shifted left by 1 unit.

• The height is stretched horizontally by a factor of 2 and shifted down by 4 units.

• The height is compressed vertically by a factor of 2 and shifted down by 4 units.

• The height is stretched vertically by a factor of 2 and shifted up by 4 units.

A farmer's crop yield can be modeled by the function Y(x) = x^3 - 6x^2 + 9x. What does the coefficient of x^2 suggest about the yield as the number of plants increases?

• The yield will decrease steadily with more plants.

• The yield will initially increase but eventually decrease as the number of plants increases.

• The yield will continuously increase as the number of plants increases.

• The yield will remain constant regardless of the number of plants.

If the function f(x) = -x^3 + 4x^2 - 5 is used to model the path of a thrown ball, what does the negative coefficient of x^3 indicate about the ball's trajectory?

• The negative coefficient means the ball will only rise and never fall.

• The ball's trajectory is a straight line, indicating it will not change height.

• The ball's trajectory is a parabola that opens upwards, indicating it will fall after reaching a peak.

• The ball's trajectory is a downward-opening curve, indicating it will rise and then fall.

A toy manufacturer finds that the cost to produce x toys is given by C(x) = 3x^3 - 15x^2 + 24x. How does the coefficient of x^3 affect the cost as production increases?

• The coefficient of x^3 makes the cost constant regardless of production levels.

• The coefficient of x^3 causes the cost to increase at an accelerating rate as production increases.

• The coefficient of x^3 has no effect on the cost as production increases.

• The coefficient of x^3 causes the cost to decrease as production increases.

The function f(x) = (x + 2)^3 - 5 represents the height of a roller coaster. What transformation has been applied to the basic cubic function?

• The function has been shifted left by 2 units and down by 5 units.

• The function has been stretched vertically by a factor of 3 and shifted left by 2 units.

• The function has been shifted right by 2 units and up by 5 units.

• The function has been reflected over the x-axis and shifted up by 5 units.

A swimming pool's volume can be modeled by the function V(x) = 4(x - 3)^3 + 10. What does the transformation indicate about the pool's dimensions?

• The pool's dimensions are shifted left by 3 units, stretched vertically by a factor of 2, and raised by 10 units.

• The pool's dimensions are shifted right by 3 units, stretched vertically by a factor of 4, and raised by 10 units.

• The pool's dimensions are shifted left by 3 units, compressed vertically by a factor of 4, and lowered by 10 units.

• The pool's dimensions are shifted right by 3 units, compressed vertically by a factor of 4, and raised by 5 units.

The function f(x) = -x^3 + 6x^2 - 9x models the profit of a business. How do the coefficients of the function affect the maximum profit point?

• The coefficients only affect the profit at the minimum point.

• The coefficients determine the total revenue instead of profit.

• The coefficients determine the shape and location of the maximum profit point.

• The coefficients have no impact on the profit function.