Quadratic equations describe parabolic shapes and can be used to model projectile motion. They take the form ax^2 + bx + c, where a cannot be 0. This document discusses using a quadratic equation, h(t) = -16t^2 + 32t + 128, to find the time when a ball thrown in the air hits the ground. By graphing the equation and finding where the quadratic intersects the x-axis at height h=0, the time is determined to be 4 seconds.

1. {
Quadratic Equations
How can we use them?

2. A quadratic is an algebraic expression of the form
ax2 + bx + c
where a, b, and c are real numbers and a is not zero.
What is a Quadratic?
Expression Is it a Quadratic? Why?
5x2 – 7x + 4 Yes a, b, and c are all real numbers
-3x2 + 8x Yes c = 0, and 0 is a real number
x3 + 2x2 – 3x + 5 No
The highest power of x must be two, so the x3
in the expression disqualifies it as a quadratic
4x + 7 No
a = 0 in this expression, which is not
permitted in quadratics
x2 Yes
a, b, and c are all real numbers;
a = 1, b = 0, and c = 0

3. A common problem we encounter that we can
use quadratics to solve involves throwing a ball.
When Do We Use Quadratics?

4. A quadratic equation forms what is called a parabola
when graphed, and it is shaped like the arc of the ball
in the picture below
When Do We Use Them?

5. If we have a quadratic equation that tracks the
height of the ball over time, we can use it to figure
out when the ball will hit the ground.
When Do We Use Them?
Where the ball hits the ground

6. Suppose the equation for a ball being thrown is
h(t) = -16t2 + 32t + 128
where h is the height of the ball, and t is the time that has passed
Solving the Problem
Here is the resulting graph:
Since we want to find when
the ball hits the ground, and
the height at ground level is
0, we need to find the time,
t, when the height, h, is 0.

7. Solving the Problem
It is clear from this graph that the time when the ball
hits the ground will be 4 seconds, and thus we have
answered our question.
The point
where the
ball hits the
ground

8. Solving the Problem
Note that there is another place where the height is 0.
We do not use this for our answer because the time, t,
would be negative at this point, which does not make
sense in the context of our problem.
Another
point where
the height is
zero.

9. Conclusion
Thus we have used the graph of a quadratic equation to
answer our question, and we can conclude that a ball that
is thrown and has the equation h(t) = -16t2 + 32t + 128 will
hit the ground after four seconds.
There are other methods we can use to solve these types
of questions, one of them being the method of factoring.
This is the method that our next lesson will address.

10. Boy Throwing Ball
http://www.ux1.eiu.edu/~cfadd/1150/08Statics/Images/ball.gif
Graph of Quadratic Equation
http://www.algebra.com/algebra/homework/equations/Equations.fa
q.question.411053.html
Images: