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NYS Algebra Regents June 2020 Q36 Explained

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NYS Algebra Regents June 2020 Q36 Explained

The New York State Algebra Regents exam is a crucial test for high school students, and question 36 is often considered one of the more challenging problems. In this blog post, we will delve into the June 2020 exam's question 36, providing a comprehensive explanation and solution strategy.

Understanding the Problem

The question presents a scenario involving a function that models the height of a ball thrown vertically into the air. The function is given as:

h(t) = -16t2 + 64t + 80

where h(t) represents the height of the ball in feet and t represents the time in seconds after the ball is thrown. The question asks us to determine the maximum height the ball reaches.

Solution Strategy

To find the maximum height, we need to recognize that the function h(t) is a quadratic function, and its graph is a parabola. The maximum height corresponds to the vertex of this parabola.

There are two common methods to find the vertex of a parabola:

  1. Completing the Square: This method involves rewriting the quadratic function in vertex form, h(t) = a(t - h)2 + k, where (h, k) represents the vertex.
  2. Using the Vertex Formula: The vertex formula directly calculates the x-coordinate (in this case, the time t) of the vertex using the formula t = -b / 2a, where a and b are the coefficients of the quadratic function.

Step-by-Step Solution

Let's solve this problem using the vertex formula:

  1. Identify the coefficients: In the function h(t) = -16t2 + 64t + 80, we have a = -16 and b = 64.
  2. Calculate the time (t) of the vertex: Using the vertex formula, t = -b / 2a, we get t = -64 / (2 * -16) = 2 seconds.
  3. Substitute the time (t) back into the function: To find the maximum height, we substitute t = 2 into the function h(t):

    h(2) = -16(2)2 + 64(2) + 80 = 144 feet.

Answer

Therefore, the maximum height the ball reaches is 144 feet.

Key Takeaways

  • Understanding the relationship between quadratic functions and their graphs (parabolas) is crucial for solving problems like this.
  • The vertex of a parabola represents the maximum or minimum point of the function, depending on the sign of the leading coefficient.
  • The vertex formula provides a quick and efficient method for finding the vertex of a parabola.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. A rocket is launched vertically into the air with an initial velocity of 128 feet per second. The height of the rocket, in feet, after t seconds is given by the function h(t) = -16t2 + 128t. What is the maximum height the rocket reaches?
  2. The profit, P(x), earned by selling x units of a product is given by the function P(x) = -0.5x2 + 100x - 500. How many units must be sold to maximize the profit?

By practicing these problems, you will gain confidence in tackling similar questions on the NYS Algebra Regents exam.

Remember, understanding the concepts and applying appropriate strategies is key to success in any exam. Good luck with your preparation!

Vertex of a Parabola