Applications of Quadratic Equations

The soccer ball

The scatter plot to the left shows $y$, the approximate height in meters of a soccer ball, $x$ seconds after it has been kicked into the air.

If you connected the points plotted to the left, would they look more like a straight line or more like a parabola?

What kind of equation do you think best models the path of the ball: a linear equation or a quadratic equation?

Slide the values of $a$, $h$, and $k$, and find an equation in the form $y=a(x-h)^2+k$ that matches the data. (Use the location of the vertex to find $h$ and $k$. Use the shape of the parabola to find the value of $a$. The error to the right of the grid will be 0.0 — and it will disappear — when your parabola matches the data.)

Click or tap inside the grid to the left and drag to the left or right. To the right of the grid you will see the $y$-value of the point on the graph with the $x$-value you have selected. You can also use the left and right arrow keys to move in small increments.

How high is the ball 1 second after it has been kicked? (Remember that the $x$-coordinate on the graph represents the time since the ball was kicked.)

meters

When is the ball 8.975 meters above the ground?

seconds and
seconds

What is the greatest height the ball reaches?

meters

Example: Find an equation for the parabola without using the sliders.

Looking at the points, you can see that the vertex of the parabola is at $(2,20)$, so in the equation $y=a(x-h)^2+k$ you can replace $h$ with 2 and $k$ with 20:

	$y=a(x-h)^2+k$
Vertex at $(2,20)$ ⇒	$y=a(x-2)^2+20$

Now all that is left to do is find the value of $a$. To do this, pick another point on the graph, say $(0,0.4)$. Because this point is on the graph, you can substitute 0 for $x$ and 0.4 for $y$ in $y=a(x-2)^2+20$:

Point $(0,0.4)$	⇒ $0.4=a(0-2)^2+20$
	⇒ $0.4=a(4)+20$
	⇒ $-19.6=4a$
	⇒ $a=-4.9$

So, an equation for the parabola is: $y=-4.9(x-2)^2+20$

$x$	$y$

The Golden Gate Bridge

Above is a picture of the Golden Gate Bridge. Its main cables have the shape of part of a parabola. Each tower of the bridge rises 152 meters above the roadbed. The length of the main span is 1280 meters. We wish to find an equation for a parabola that could model the bridge’s main cables. The diagram below shows a graph of the main cables of the Golden Gate Bridge.

Golden Gate Bridge

The vertex of the parabola is assumed to be at the origin. Since it is halfway between the two towers, the distance between the vertex and the towers is $$1280/2$$ meters or 640 meters. Because the towers are 152 meters tall, this means the points $(-640, 152)$ and $(640, 152)$ must be on the parabola.

Use the sliders to find an equation in the form $y=a(x-h)^2+k$ that approximates the main cables of the Golden Gate Bridge to within about 0.5 meters.

Use the method shown in the example below findHeight to find an equation for the main cables of the Golden Gate Bridge.

Since the vertex of the parabola is at $(0,0)$, what form does the equation take? (What are $h$ and $k$?)

Solve for $a$, by using the fact that $(640,152)$ satisfies that equation. Write your final answer as a decimal, rounded to as many decimal places as you can fit in the answer box. (You can use a calculator to do the arithmetic.)

What equation does this give for the cables?

Baseball

One of the most important discoveries in science was the description of how gravity affects objects rising from or falling to the earth’s surface. In 1638 Galileo claimed that the height of such an object is a quadratic function of its time in the air.

Suppose that a baseball player hits the ball straight above home plate. If the bat meets the ball 0.49 meters above the ground and sends it up at a rate of 30.42 meters per second, then the height of the ball, in meters, $t$ seconds later is predicted by the rule:

$$h=-4.9t^2+30.42t+0.49$$

Click or tap inside the grid to the left and drag to the left or right. To the right of the grid you will see the $h$-value of the point on the graph with the $t$-value you have selected. You can also enter a value for $t$ directly into its input box. Use the graph and the equation to answer the following questions.

What is the height of the ball 2 seconds after it has been hit?

meters

Approximately when is the ball more than 40 meters above the ground?

Does the ball ever get more than 50 meters above the ground?

Remember, the baseball player hit the ball so that it traveled up at a rate of 30.42 m/s. Can you find the number 30.42 in the equation for $h$ (that is, in $h=-4.9t^2+30.42t+0.49$)?

Is 30.42 the quadratic coefficient, the linear coefficient, or the constant coefficient of that equation?

What do you think an equation for the height of the ball would be if the bat gave it a vertical velocity of 27 meters per second?

Use the slider or input box for $b$ to change the equation to the equation you found in the last question ($h=-4.9t^2+27t+0.49$). Approximately what is the maximum height the ball will reach in this case?

meters

Approximately what is the maximum height of the ball if the bat gives it a vertical velocity of 36.9 meters per second?

meters

How long will the ball be in the air if it’s given a vertical velocity of 13.05 meters per second?

seconds

Maximizing profit

A bakery sells more loaves of bread when it reduces its price, but if the price is too low, then the profits are very low. The equation

$$p=-100x^2+360x-175$$

models the bakery’s daily profits in dollars, where $x$ is the price of a loaf of bread in dollars. To the left is a graph of this equation.

What is the profit when selling the bread at \$2.10 per loaf?

What is the profit when selling the bread at \$1.20 per loaf?

What price should the bakery charge to maximize its profits?

\$ per loaf

What is the maximum profit?