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Parabolas
A U-shaped graph is called a parabola. It can open up or down as shown here:
Definition: A parabola is the set of all points equidistant from a fixed point and a fixed line.
The fixed point is called the focus.
The fixed line is called the directrix.
The vertex is the turning point of the parabola.
What type of equation will make a parabola when you graph it? A quadratic equation!
"Quadratic" just means that the x term is squared.
There are two forms of quadratic equations:
Standard form: y = ax2 + bx + c (a,b,c are numbers)
Examples: y = 2x2 + 3x – 1
y = 4x2 - 5
2
y = x -5x
y = -6x2
Vertex form: y = a(x-h)2 + k (This is the form we will be using for this lesson )
Examples:
y = 3(x-2)2 -1
y= -(x+4)2
y= -2(x-5)2
y= x2 – 8
a=3 h=2 k= -1
a= -1 h= -4 (since the form has x MINUS h)
k=0 (since there is no number added or subtracted at the end)
a = -2 h=5 k=0
a=1 (since there is no number in front), h=0 since there is no
number in parentheses with the x
k= - 8
*Note: If "a" is positive (a>0), the parabola will open up.
If "a" is negative (a<0), then the parabola will open down.
How do you find the vertex,focus and directrix?
Vertex form: y = a(x-h)2 + k
The point (h,k) is the vertex.
The distance (d) from the vertex to the focus is
.
Example: Find the vertex,focus and directrix for y = 2(x-1)2+5
y = 2(x-1)2+5 is like the vertex form y = a(x-h)2 + k. so a = 2, h = 1, and k = 5
The vertex is (h,k) which is (1,5)
To find the vertex and directix, we need to find d, the distance from the vertex to the focus or directrix.
The formula is
1
1
1


d
4a 4(2) 8
Graph the vertex and a sketch of what it will look like.
Since a = 2 which is greater than 0, we know it opens up.
The focus is 1/8 of a unit UP from the
parabola.
When you move UP, you ADD to the y
coordinate.
The vertex is (1,5) so the focus is
(1, 5+1/8) which is (1, 5+.125)
Focus: (1,5.125)
The directrix is a horizontal line so the
equation is always y = ???
The directrix is 1/8 of a unit DOWN from
the vertex.
So take the y-coordinate and
SUBTRACT 1/8 (or .125)
Y = 5 – 0.125
Directrix: y = 4.875
If the equation is not in vertex form, you will have to complete the square.
Example: Find the vertex of y = x2 - 4x+3
We will have to complete the square on this one to get it into vertex form.
y = x2 - 4x+3
y = (x2 - 4x
)+3
To complete the square, take HALF of the 4 and then square it.
½ * 4 = 2 and then 2 squared = 4
So add 4 to the inside of the parentheses and then do the OPPOSITE and subtract it at the end.
y = (x2 - 4x +4 )+3 -4
y = (x2 - 4x +4 ) -1
y = (x-2)2 -1
The vertex is (2, -1)
Example: Find the vertex, focus and directrix of y= - x2 +6x -1
We need to complete the square here. Remember that the x2 term can not have the negative in front
so factor it out.
y= - (x2 -6x ) -1
y= - (x2 -6x +9 ) -1 +9
Take half of the 6 and square it so ½ * 6 = 3 and 3 squared is 9.
Remember to distribute the negative in front of the parentheses so you
really SUBTRACTED 9 in parentheses. You need to ADD 9 at the end.
y= - (x2 -6x +9 ) +8
y= - (x2 -3) 2 +8
The vertex is (3, 8)
d = 1/4a = 1 / 4*(-1) = -1/4
Graph the vertex at (3,8). The graph will open down so the focus will be BELOW the vertex and the
directrix will be ABOVE it d = ¼ units.
Focus = (3,8 – ¼ ) = (3,7.75)
Directrix : y = 8 + ¼ = 8+0.25 so y = 8.25
Example: Find the vertex, focus and directrix of y=2x2 -4x +3
Again, we have to complete the square.
WARNING: Remember you have to factor out the 2 from the first two terms before you start.
y=2x2 -2x +3
y =2(x2 -2x )+3
Take half of the -2 and then square it to figure out what to add inside the parentheses.
½ * -2 = -1 then square that to get +1
y=2(x2 -2x +1) +3 – 2 (remember you have to distribute the 2 in front to the 1 = 2 * 1 = 2)
y = 2(x2 -2x +1) +1
y = 2(x2 -1) 2 +1
The vertex is (1, 1)
The distance d from the vertex to the focus is 1 / 4*2 = 1/8.
Since a = +2, the parabola opens UP so the focus is 1/8 of a unit UP from the vertex.
Focus = (1, 1 + 1/8) = (1, 1.125)
The directrix is 1/8 of a unit DOWN from the vertex.
Directrix: y = 1 – 1/8 =7/8 so y = 7/8 or y =0.875
Example: Write the equation of the parabola with focus (2,3) and directrix y = -1.
Steps
1. Find vertex by sketching a graph.
2. Find the distance from focus to vertex by looking at graph. That distance is d.
3. Fill in equation. y=a(x-h)2+k
Vertex = (h,k) and a = 1
4d
The distance between the focus and
the directrix is 4 units.
The vertex must be halfway
between those two points so from
the focus go DOWN 2 units.
The vertex is in the middle at (2,1).
Since the vertex is (h,k) then
h=2 and k = 1.
d =distance between VERTEX and
focus so d = 2 here
a = 1/(4d) = 1/(4 * 2) = 1/8
Fill in the equation
y=a(x-h)2+k
y=1/8(x-2)2+1
Last example: Write the equation of the parabola with focus (-1,3) and directrix y = 2.
Graph the focus and the directrix as shown below.
The vertex must be in the middle between the point and the line so it must be at (-1,2+1/2) so (-1,2.5)
That means that (h,k) = (-1,2.5) so h = -1 and k = 2.5
To find a, remember the formula a = 1/(4d)
d is the distance between the VERTEX and the focus as shown here
d = the distance between (-1, 3.5) and (-1.3)
so d = 0.5
a = 1/(4d) = 1/(4*0.5) = ½
Equation:
y=a(x-h)2+k
y= ½ (x+1)2+2.5
Transformations with Graphs of Parabolas
How does changing the numbers of a quadratic function affect the graph?
If the graph is in vertex form: y=a(x-h)2+k, then
Changing the "a" makes the parabola wider or narrower (wider if a<1 and narrower if a>1)
Changing the "h" shifts the graph left or right OPPOSITE the sign (+ moves left and - moves right)
Changing the "k" shifts the graph up or down (+ is up and - is down)