Use quadratic equations to figure out solutions to real-life situations
Why study quadratic functions?
An arrow is shot into the air. How high will it go? How far away will it land? It turns out that you can answer these and related questions with just a little knowledge of quadratic functions. In fact quadratic functions can be used to track to the position of any object that has been thrown, shot, or launched near the surface of the Earth. As long as wind resistance does not play a huge role and the distances are not too great, you can use a quadratic function to model the flight path.
Suppose an archer fires an arrow from a height of [latex]2[/latex] meters above sea level on a calm day. While the arrow is in the air, someone else tracks and records its height precisely at each second.
The following table shows the arrow’s height (meters) versus time (seconds).
Time ([latex]t[/latex])
Height ([latex]h[/latex])
[latex]0[/latex] sec.
[latex]2[/latex] m
[latex]1[/latex] sec.
[latex]16.7[/latex] m
[latex]2[/latex] sec.
[latex]21.6[/latex] m
Let’s plot the data on a coordinate plane.
Notice the up-and-down shape? As Isaac Newton would say, what goes up must come down. The smooth arc, first rising and then falling, is a tell-tale clue that there is a quadratic function lurking in the data. The curve that best fits this situation is a parabola, which is what we call the graph of a quadratic function. With a little more work, you can find the equation of this function:
[latex]h(t)=-4.9t^2+19.6t+2[/latex]
In the above equation, [latex]t[/latex] represents time in seconds, and [latex]h[/latex] represents height in meters.
Let’s analyze this quadratic function!
The vertex of the parabola represents the maximum height of the arrow since the leading coefficient, [latex]a = -4.9[/latex], is negative (indicating a downward opening parabola).
From the table, we can see that the vertex is [latex](2, 21.6)[/latex].
We can also find the vertex using the quadratic function:
First, find the [latex]x[/latex]-coordinate of the vertex using the formula [latex]t = -\frac{b}{2a}[/latex].
[latex]\begin{align*} t &= -\frac{19.6}{2 \cdot -4.9} \\[1.5mm] t &= -\frac{19.6}{-9.8} \\[1.5mm] t &= 2 \text{ seconds} \end{align*}[/latex]
Next, find the [latex]y[/latex]-coordinate of the vertex by substituting [latex]t = 2[/latex] back into the function:
Interpreting the Vertex: In the context of the problem, the vertex provides two key pieces of information:
Maximum Height: The [latex]y[/latex]-coordinate of the vertex ([latex]21.6[/latex] meters) represents the maximum height the arrow reaches.
Time to Reach Maximum Height: The [latex]x[/latex]-coordinate of the vertex ([latex]2[/latex] seconds) represents the time it takes for the arrow to reach this maximum height.
To find the time when the arrow hits the ground, set [latex]h(t) = 0[/latex] and solve for [latex]t[/latex]. We can use the quadratic formula using [latex]a = -4.9[/latex], [latex]b = 19.2[/latex], and [latex]c = 2[/latex] to find this answer.
So, the [latex]x[/latex]-intercepts are approximately [latex]t = -0.09796[/latex] (which is negligible before the launch) and [latex]t = 4.09[/latex] seconds.
Interpretation of the Roots: The positive root value [latex]t = 4.09[/latex] seconds represents the time when the arrow hits the ground after being fired. Specifically, it indicates that the arrow reaches the ground approximately [latex]4.09[/latex] seconds after it is launched.
To find the [latex]y[/latex]-intercept of a quadratic function, set [latex]t = 0[/latex] and solve for [latex]h(t)[/latex].
So, the [latex]y[/latex]-intercept is [latex](0,2)[/latex].
Interpretation of the [latex]y[/latex]-intercept: In the context of the problem, the y-intercept represents the initial height of the arrow. So, the height from which the arrow is fired is [latex]2[/latex] meters.
Quadratic functions are a fundamental concept in algebra and mathematics. They can model various real-world phenomena, such as the trajectory of projectiles, the shape of parabolic structures, and more.
Vertex: The vertex of a parabola is the highest or lowest point, depending on the direction the parabola opens.
Roots: Points where the graph of the function crosses the [latex]x[/latex]-axis.
[latex]y[/latex]-intercept: The point where the graph of the function crosses the [latex]y[/latex]-axis. In application problems, it is typically the initial value.
