**Lesson 4:**

**Parabolas**

**By Carly
Coffman**

**Or**

**Definitions: **

Ø
**A parabola is the set of
all points (x,y) in a plane that are equidistant from a fixed line, the
directrix, and a fixed point, the focus, not on the line.**

Ø
**The vertex is the midpoint
between the focus and the directrix.**

Ø
**The axis of the parabola
(or axis of symmetry) is the line passing through the focus and the vertex.**

**Look at the creation of a
parabola (you must have GSP to see this link):
Parabola**

**Let’s explore the parabola
a bit. In the equation, ****, let’s
look at the affect h and k have on the graph.**

**1) Do you have any predictions as to what increasing h and k will do
to the graph?**

** *Remember to respond in complete sentences.**

**Let’s see if you were on
target or not. Click on the play button
at the bottom of the window. Exploration 1**

**2)
****What is the relationship
between h and the parabola graph?**

**Now, let’s test what k
does to the graph. Click on the play
button at the bottom of the window. Exploration 2**

**3)
****What is the relationship
between k and the parabola graph?**

**Now you should be able to
tell where the vertex of each of the following parabolas lies. State the vertex of each parabola in a
complete sentence. Copy and paste the equation into your Word document.**

** **

**4)
**

**5)
**

**Notice also that all of
the equations we have worked with so far have had the x term squared. Each one of these graphs has been concave up,
which means the vertex is the minimum point on the y-axis and the graph looks
like a smiley face.**

**6)
****What do you think would
happen if there was a negative sign in front of the parenthesis of the
x-term? **

**Let’s see if your
prediction was correct. Exploration 3 **

**7)
****Was your prediction
correct? How did the negative sign
affect the graph?**

**When the vertex is the
maximum point on the graph, the graph is concave down. These parabolas look like a frown.**

**Let’s see what happens
when the y-term is squared. **

**State the vertex and axis
of symmetry of the following parabolas.
Remember the axis of symmetry is the equation of the line of symmetry
for each parabola.**

** 8) **

** **

** 9) **

**10)
****What do you predict will
happen to the parabolas above when a negative sign is placed in front of the
parenthesis for the y-term? Test your
prediction and tell whether or not your prediction was correct.**

**Now, we have one last
aspect of parabolas to explore. The last
aspect is the value of p in each parabola equation. In most of the equations
above, p has been one. **

**11)
****What do you think p
affects on the graph of a parabola? (take a guess if you do not know)**

**Remember, the coefficient
of x is 4p. So, in order to get the
value of p we must divide the coefficient of x by 4. The first p is 2, the second p is ¼ and the
third p is 1/16. **

** **

**12)
****As p decreases, what
happens to the graph of the parabola?**

**We have finished exploring
the variables of parabolic equations. Print
your Word document and file it with your other conic explorations.**

**Congratulations
you have finished your parabola exploration!**

**Return to Home Page Next
Lesson**

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