Problem 60 - Algebra and Trigonometry

Question

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x),\) in feet, can be modeled by $$ f(x)=-0.8 x^{2}+3.2 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

Step-by-Step Solution

Answer

a) The maximum height of the ball is 4 feet and it occurs 2 feet from where it was thrown. b) The ball travels 5 feet horizontally before hitting the ground. c) The graph of the function is a downward-opening parabola with a vertex at (2, 4), beginning at (0, 6), and ending at (5, 0)

Step 1: Finding the maximum height

The maximum height of the ball is at the vertex of the parabola. For a parabola given by the equation \(f(x) = ax^2 + bx + c\) the x-coordinate of the vertex is given by \(-b / 2a\). For our equation, \(a = -0.8\) and \(b = 3.2\), so the x-coordinate of the vertex is at \(-3.2 / (2*-0.8) = 2\). The y-coordinate (or the height) at this point is \(f(2) = -0.8(2^2) + 3.2(2) + 6 = 4\) feet, which is the maximum height.

Step 2: Finding the horizontal distance before the ball hits the ground

The ball hits the ground when its height, \(f(x)\), is zero. So we need to solve the equation \(-0.8x^2 + 3.2x + 6 = 0\). Factoring out \(-0.8\) gives us \(0.8x^2 - 4x - 7.5 = 0\). Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find that the roots are \(x = 1.875\) and \(x = 5\). Since we're considering distance, we ignore the negative root, and thus the ball travels 5 feet horizontally before hitting the ground, to the nearest tenth of a foot

Step 3: Graphing the function

To establish the graph of the function, plot a few points. Begin with the vertex at (2, 4), the initial point at (0, 6), the final point at (5, 0), and perhaps a few others in-between as needed. Then sketch the parabola that passes through these points.

Key Concepts

Quadratic Functions

When we speak of quadratic functions, we're delving into the realm of algebra where variables are raised to the second power. The general form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. This distinct form creates a U-shaped graph known as a parabola.

Understanding quadratic functions is essential not just in mathematics, but in various real-world applications. For example, the motion of objects thrown in the air, such as a ball, often follows a quadratic function. The coefficients of the equation affect the parabola's width, direction, and location on the graph. If \( a > 0 \), the parabola opens upwards, and conversely, if \( a < 0 \), it opens downward.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on its graph, depending on whether the parabola opens upward or downward, respectively. For our quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). This x-value is then plugged back into the function to obtain the y-value, giving us the coordinates of the vertex.

The vertex is a crucial aspect in understanding the parabola's shape and the outcome of real-world problems like calculating the maximum height of a ball thrown in the air. To convey this concept effectively, it's helpful to visualize or plot the vertex on a graph, showing how it relates to the overall shape of the parabola.

Graphing Parabolas

Graphing parabolas is a visual representation of quadratic functions. It allows us to see the relationship between the quadratic equation and its graphical counterpart. To graph a parabola, we first find key points such as the vertex, y-intercept, and where the parabola intersects the x-axis (if it does).

Typically, we start by plotting the vertex, followed by the y-intercept which is the point \( (0, c) \). Additional points are then determined by selecting x-values and calculating their corresponding y-values. Once enough points are plotted, we draw a smooth curve through them, bearing in mind the parabola's symmetry about the vertical line that passes through the vertex. Graphing tools, both physical and digital, streamline this process, aiding in a more precise and clearer understanding of the function's behavior.

Quadratic Formula

The quadratic formula is the solution to the quadratic equation \( ax^2 + bx + c = 0 \). The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives us the values of x that make the quadratic equation true. These values are called the roots or solutions of the equation.

The discriminant, \( b^2 - 4ac \), in the quadratic formula is very telling; it indicates whether the roots are real and distinct, real and the same, or complex. For instance, if the discriminant is positive, we have two real and distinct roots. If it's zero, the roots are real and identical. A negative discriminant means the equation has complex roots. The quadratic formula is a powerful tool used to find where a parabola intersects the x-axis or to solve real-world problems involving parabolic motion.

Horizontal Distance Calculation

Calculating the horizontal distance an object travels while in the air in a parabolic motion is an important application of quadratic functions. When an object is projected forward, its height as a function of horizontal distance often forms a parabola.

To find the total horizontal distance, we set the height \( f(x) \) to zero and solve for \( x \). This gives us the distance from the starting point to where the object lands. In practice, like for a ball thrown in the air, we only consider the positive root of the equation since distance cannot be negative. By employing the quadratic formula, we can solve for this distance even if the quadratic equation is not easily factorable. This principle is crucial in fields such as physics and engineering, where projectile motion is regularly analyzed.