1. Describing the graph of a Parabola
There are four different ways to describe the graph of a parabola:
It intersects the x-axis twice.
It is tangent to the x-axis. (It only intersects it once)
It lies entirely above the x-axis.
It lies entirely below the x-axis.
We are going to explore all of these ways by looking at the discriminant and the coefficient of the squared term.

2. We need to be able to determine if a parabola opens up or down
We need to be able to determine if a parabola opens up or down. To do that, we look at the coefficient of the squared term.
Now type the following equation into y= on your calculator.
Type in the following equation into y= on your calculator.
What did we learn?
If the squared term is positive, it opens up. If it is negative, it opens down!

3. We need to be able to determine what makes a parabola touch the x-axis and lie above or below it:
The discriminant is a positive, non-perfect square. Therefore the roots are:
1. Real
2. Irrational
3. Unequal
If the roots are real, the parabola touches the x-axis:
Ex:
We would describe this graph as touching the x-axis twice!

4. If the roots are imaginary, the parabola doesn’t touch the x-axis:
Opens down
The discriminant is negative. Therefore the roots are:
1. Imaginary
We would describe this graph as follows:
It lies entirely below the x-axis.
If the roots are imaginary and the coefficient of the squared term is negative, it lies entirely below the x-axis. If the roots are imaginary and the squared term is positive, it lies entirely above the x-axis.

5. Page 2
Now lets determine what the graph looks like.
#16:
#13:
Parabola opens up
Parabola opens up
The discriminant is a positive, non-perfect square. Therefore the roots are:
1. Real
2. Irrational
3. Unequal
The discriminant is a positive, perfect square. Therefore the roots are:
1. Real
2. Rational
3. Unequal
Since the roots are unequal and real, the graph intersects the x-axis twice!
Since the roots are unequal and real, the graph intersects the x-axis twice!

6. Page 2
#19:
#20:
Parabola opens up
Parabola opens down
The discriminant is zero. Therefore the roots are:
1. Real
2. Rational
3. Equal
The discriminant is negative. Therefore the roots are:
1. Imaginary
Since the roots are equal the parabola is tangent to the x-axis.
Since the roots are imaginary and the parabola opens down, then the parabola lies entirely below the x-axis.

7. Page 2
#34: The roots of are equal when k is:
Remember, the roots are equal when the discriminant is 0.

8. Page 2
#36: The roots of are imaginary when b is:
Remember, the roots are imaginary when the discriminant is negative.
< 0 means negative!
Since this is multiple choice, sub in values from choices until one works!

9. Page 2
#38: If the graph of is tangent to the x-axis, then the roots of are:
If a graph is tangent to the x-axis, it only touches the x-axis__________.
ONCE
The roots are:
Real
Rational
Equal

10. Page 2
#36: The roots of are equal when b is:
Remember, the roots are equal when the discriminant is zero.
= 0 means equal roots.

11. Find the largest integral value of k such that the roots of the given equation are real.
means real.

12. Homework
Page 2
#15,18,21,35,41,
43,46