Question

A baseball is thrown up in the air. The table shows the heights y (in feet) of the baseball after x seconds. Write an equation for the path of the baseball. Find the height of the baseball after 5 seconds $$ \begin{array}{|l|c|c|c|c|} \hline \text { Time, } \boldsymbol{x} & 0 & 2 & 4 & 6 \\ \hline \text { Baseball height, } \boldsymbol{y} & 6 & 22 & 22 & 6 \\ \hline \end{array} $$

Short answer

The equation of the path of the baseball is \(y = -2x^2 + 8x + 6\) and the estimated height of the baseball after 5 seconds is 1 foot.

Step-by-step solution

Find the parameters of the equation

Taking three points from the table \((0, 6)\), \((2, 22)\), and \((4, 22)\), for example, replace \(x\) and \(y\) in equation \(y = ax^2 + bx + c\), to get three linear equations, and solve for \(a\), \(b\), and \(c\). You should get \(a = -2\), \(b = 8\), and \(c = 6\) to satisfy these three points.

Construct the equation

Replace \(a = -2\), \(b = 8\), and \(c = 6\) in the quadratic equation \(y = ax^2 + bx + c\) to obtain \(y = -2x^2 + 8x + 6\). This is the path of the baseball.

Solve for the height after 5 seconds

Now, find the height of the baseball after 5 seconds, by replacing \(x = 5\) into the equation. Solve for \(y\) to find the height of the baseball at that particular time. Doing this, you should get \(y = -2(5^2) + 8*5 + 6 = 1.\)

Key concepts

Parabolic Motion

Parabolic motion is a type of movement observed when an object is projected into the air and follows a curved path. This motion is called 'parabolic' due to the parabolic shape of its trajectory, which is a result of the gravitational force acting on the object.
When a baseball is thrown up in the air, it moves upward against gravity, eventually reaches a peak, and then descends. This path forms a parabola, with one vertex at the highest point of the motion.
Understanding parabolic motion involves recognizing how different factors, like angle and speed of projection, affect the object's height and distance traveled. In physics, this motion can be analyzed by using quadratic equations, which allows us to predict the object's position at any point in time.

• The path is symmetrical, meaning the ascent and descent times are equal.
• The maximum height is reached when the upward and downward forces balance, temporarily.

Quadratic Functions

Quadratic functions are expressions of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. They are crucial in describing parabolic motion.
Here, \(a\) influences the direction and width of the parabola. If \(a\) is negative, like in our baseball example, the parabola opens downwards, forming a concave shape. With a positive \(a\), the parabola opens upwards.

• The vertex of the parabola represents either the maximum or minimum value.
• The 'axis of symmetry' is a vertical line passing through the vertex, splitting the parabola into two mirrored halves.
• The roots or solutions of the quadratic function indicate the points where the parabola intersects the x-axis.

Using quadratic functions, we can describe complex motions and predict variables like time, height, or range in these scenarios.

Solving Quadratic Equations

Solving quadratic equations can often be done through several methods. These include factoring, completing the square, or using the quadratic formula. Each method offers a tool to find the roots, or x-values, where the equation equals zero,
In solving our exercise, we applied values directly to the quadratic form to find parameters like \(a = -2\), \(b = 8\), and \(c = 6\). Once the specific equation is determined, we can plug in other values, such as \(x = 5\), to predict results for different time points,

• Factoring is useful when equations are easily separable into terms such as \((x-p)(x-q) = 0\).
• The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides solutions when other methods are not straightforward.
• Completing the square helps derive the function into the vertex form \(y = a(x-h)^2 + k\), offering insights into the vertex's position.

By understanding these concepts, students can efficiently analyze and solve quadratic equations, applicable in a range of mathematical and real-world problems.