Transformations of Functions

Consider the trajectory of a ball thrown into the air. The height of the ball at any time t is given by the equation: h(t) = 3t^2 + 2t - 5. Determine the nature of this equation.

• exponential

• quadratic

• cubic

• linear

Consider a company where the profit, P(x), in thousands of dollars, is a linear function of the number of products sold, x, given by P(x) = 2x - 3. What is the slope of this profit line?

• 0

• 2

• 5

• -3

Imagine a theme park ride's path is modeled by the function h(x) = (x-2)^2 - 4, representing the height of the ride above ground level in meters, where x is the time in seconds since the ride started. How does this path compare to the standard path modeled by the function y = x^2?

• The path of the ride compared to the standard path is a reflection over the time axis and a reflection over the height axis.

• The path of the ride compared to the standard path is a horizontal shift of 2 seconds to the right and a vertical shift of 4 meters downward.

• The path of the ride compared to the standard path is a vertical stretch by a factor of 2 and a horizontal compression by a factor of 4.

• The path of the ride compared to the standard path is a vertical shift of 2 meters upward and a horizontal shift of 4 seconds to the left.

Imagine a roller coaster's path is described by the equation y = 2x^2 - 3. If the entire roller coaster is moved 5 units higher, what is the new equation describing its path?

• y = 2x^2 - 8

• y = 2x^2 + 5

• y = 2x^2 - 3x

• y = 2x^2 + 2

Imagine a roller coaster track designed based on the function f(x) = -2x^2, representing the steepness and curvature of the track. If the track design is mirrored over the ground level to create a new ride, what would be the equation of the new track?

• f(x) = 2x^2

• f(x) = -2x^2

• f(x) = -2x

• f(x) = x^2

Imagine a new roller coaster is being designed, and its main drop is modeled by the function g(x) = 0.5x^2, representing the height of the drop at different points. How does the steepness of this drop compare to a standard roller coaster drop modeled by the function y = x^2?

• 0.5

• 1

• 0.25

• 2

Imagine you're plotting the growth of a plant over time, and the growth can be modeled by the function f(x) = x^2. If you started measuring the growth 3 units of time later than originally planned, what would be the new equation modeling the plant's growth?

• f(x) = (x-6)^2

• f(x) = (x+3)^2

• f(x) = (x-2)^2

• f(x) = (x-3)^2

Consider a company where the profit in millions of dollars, P, can be modeled by the linear function P = -3x + 4, with x representing the number of years since the company's inception. What is the initial profit (in millions of dollars) when x=0?

• 0

• 4

• -4

• 3

Consider the scenario where a taxi company charges a base fare of $7 plus $4 per mile traveled. Identify the type of function that represents the total fare as a function of miles traveled: g(x) = 4x + 7

• exponential

• sinusoidal

• linear

• cubic

A theme park is designing a new roller coaster. The path of the roller coaster is modeled by the equation y = x^2 - 1. The design team decides to shift the entire path 2 units to the left. What is the new equation of the path?

• y = (x + 1)^2 - 1

• y = (x + 2)^2 - 1

• y = (x - 2)^2 - 1

• y = (x + 3)^2 - 1