{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.

c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)

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The height function is given by \( h = f(t) = 64t - 16t^2 \).

To find the inverse, set \( h = 64t - 16t^2 \).

Rearrange the equation to solve for \( t \) in terms of \( h \). This involves solving the quadratic equation \( 16t^2 - 64t + h = 0 \).

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 16 \), \( b = -64 \), and \( c = h \).

Since we are interested in the time when the ball is traveling downward, choose the solution with the positive square root, which corresponds to the time after the ball reaches its peak.>

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주요 개념

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Quadratic Functions

The height function h(t) = 64t - 16t² is a quadratic function, which is characterized by its parabolic shape. Quadratic functions can be expressed in the standard form ax² + bx + c, where a, b, and c are constants. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and roots, is essential for analyzing their behavior, including finding inverse functions.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(t), its inverse f⁻¹(h) allows us to find the input t for a given output h. To find the inverse of a function, one typically swaps the dependent and independent variables and solves for the new dependent variable. This concept is crucial for determining the time t at which the baseball reaches a specific height h.

Solving Quadratic Equations

To find the inverse function in this context, one must solve the quadratic equation derived from the height function. This often involves rearranging the equation to isolate t, which may require using the quadratic formula, t = (-b ± √(b² - 4ac)) / 2a. Understanding how to manipulate and solve quadratic equations is vital for accurately determining the time at which the baseball reaches a given height as it travels downward.