Applications of Quadratic Equations

The soccer ball

Question 1. 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. What kind of equation do you think best models the path of the ball: a linear equation or a quadratic equation? Why?

Question 2. 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. [Hint: 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.]

$y=$$\,(\,$$\,)^2$ 

Question 3. 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 x and $y$-values 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? meters
When is the ball 8.975 meters above the ground?
seconds &
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 a 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$
00.4
0.58.975
115.1
1.518.775
220
2.518.775
315.1
3.58.975
40.4

The Golden Gate Bridge

Question 4. The main cables of the Golden Gate Bridge have the shape of part of a parabola. Each tower of the Golden Gate Bridge rises 152m above the roadbed. The length of the main span is 1280m. We wish to find an equation for a parabola that could model the Golden Gate's main cables. The diagram below shows a graph of the main cables of the Golden Gate Bridge. The vertex of the parabola is assumed to be at the origin.

Golden Gate Bridge

How did we determine that the points $(-640,152)$ and $(640,152)$ are on the graph? (Explain where the numbers 640, $-640$ and 152 come from.)

Question 5. 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.

$y=$$\,(\,$$\,)^2$ 

Question 6. Use the method shown in the example below question 3 to find an equation for the main cables of the Golden Gate Bridge. (You can use a calculator to do the arithmetic.)

$y=$$\,(\,$$\,)^2$ 

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.

Question 7. 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. On the right hand side of the grid you can see the $t$ and $h$ values of the point you have selected. 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? $<t<$
When is the ball less than 50 meters above the ground?
Question 8. Can you find the number 30.42 in the equation for $h$ ($h=-4.9t^2+30.42t+0.49$)? Where?

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?

$h=$ $^2$

Click on Use the slider to change the equation to your answer to the last question. 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

Question 9. 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 function

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

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 function.

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? \$
What is the maximum profit? \$