Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Refer to Exercise \(37 .\) Suppose that the initial position of the frog is \((0,4)\) and its landing position is \((6,0) .\) The height of the frog in feet is given by $$h(x)=-\frac{1}{3} x^{2}+\frac{4}{3} x+4$$ (a) What was the horizontal distance \(x\) from the base of the stump when the frog reached maximum height? (b) What was the maximum height?

Step-by-Step Solution

Answer

(a) The horizontal distance is 2 feet. (b) The maximum height is approximately 5.33 feet.

Step 1: Identify the quadratic equation

The height of the frog is given by the quadratic equation: \( h(x) = -\frac{1}{3}x^2 + \frac{4}{3}x + 4 \). This is in the form \( ax^2 + bx + c \), where \( a = -\frac{1}{3} \), \( b = \frac{4}{3} \), and \( c = 4 \).

Step 2: Determine the horizontal distance at maximum height

For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex (maximum or minimum point) is given by the formula \( x = -\frac{b}{2a} \). Substituting the values of \( a \) and \( b \) from the equation, we get: \( x = -\frac{\frac{4}{3}}{2(-\frac{1}{3})} = \frac{2}{1} = 2 \). Therefore, the horizontal distance \( x \) from the base when the maximum height is reached is 2 feet.

Step 3: Calculate the maximum height

To find the maximum height, substitute \( x = 2 \) back into the height equation. \( h(2) = -\frac{1}{3}(2)^2 + \frac{4}{3}(2) + 4 = -\frac{1}{3}(4) + \frac{8}{3} + 4 \). Simplify this to get \( h(2) = -\frac{4}{3} + \frac{8}{3} + 4 = \frac{4}{3} + 4 = \frac{4}{3} + \frac{12}{3} = \frac{16}{3} \), which is approximately 5.33 feet.

Key Concepts

Maximum Height

In the world of quadratic functions, finding the maximum height of an object can be a fascinating activity. Imagine the graceful arc of a frog as it leaps, reaching a peak height before descending. This peak point reveals the frog's maximum height. In mathematical terms, a quadratic function such as \(-\frac{1}{3}x^2 + \frac{4}{3}x + 4\) represents a parabola. To determine how high the parabola goes, we need to find the point where the frog's vertical position is at its maximum. This is achieved by evaluating the quadratic function at the x-coordinate of its vertex. After calculating that x is 2, substituting this back into the function gives us the maximum height. Here, by replacing \(x = 2\) in the equation, we get a height of \(h(2) = \frac{16}{3}\), or approximately 5.33 feet. Now we can visualize the frog, positioned at its zenith, an impressive 5.33 feet above the ground. This understanding is crucial for modeling real-world scenarios where predicting maximum height is important, from engineering applications to sports science.

Vertex of a Parabola

The vertex of a parabola is the turning point, where the curve changes direction. For a "smiling" parabola (opening upwards), the vertex is the lowest point. Conversely, for a "frowning" parabola (opening downwards), like the one in our frog's leap scenario, the vertex represents the highest point. In quadratic equations of the form \(ax^2 + bx + c\), the vertex's x-coordinate is found using \(x = -\frac{b}{2a}\). This formula helps us quickly find the horizontal location of the vertex, without needing trial and error. In the frog example, identifying the vertex involves calculating \(x = -\frac{\frac{4}{3}}{2(-\frac{1}{3})}\), simplifying to 2. This indicates that the vertex, hence the maximum point of our parabola, is directly above \(x = 2\) feet from the starting point. Knowing the vertex is essential for making further predictions about an object's trajectory, like where its path might intersect other lines or objects. Grasping the concept of the vertex helps in fields such as architecture or physics. It provides insights into structural design or the physics of motion. As the nucleus of the parabola, the vertex is a key concept.

Quadratic Equation Solutions

Quadratic equations like \(ax^2 + bx + c = 0\) have wide-ranging applications, including the determination of maximum heights, turning points, and points of intersection. Solving these equations can reveal vital details about the trajectory of objects or the shape of paths. The solutions to a quadratic equation, typically known as "roots," can be calculated using methods like factoring, completing the square, or the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). These solutions indicate where the parabola crosses or touches the x-axis. However, in the context of our frog's height, the focus is on the vertex. Understanding how to manipulate and solve quadratic equations allows for precise modeling of lifelike phenomena. Whether plotting the rise and fall of rockets or predicting the path of a bouncing ball, mastering these equations helps in designing clever solutions to complex problems.