Properties of Parabolas We Parabol as puh-rab-uh- luh
Properties of Parabolas
We
Parabolas
puh-rab-uh-luh
Axis of symmetry Axis of symmetry
1. Vertex- the highest or lowest point on a parabola2. Axis of Symmetry- line that divides a parabola into 2
parts that are mirror images
3. In the figures below, label the vertex and draw the axis of symmetry
vertex
x
y
vertex
The function that models a parabola (the equation of a parabola) with its
vertex at the origin, (0,0), is Parabolas can be
skinny or fat and they can shift left or right and up or down
y=ax²
See what happens when we put in the following numbers for “a”. Use your graphing calculator
1. y=5x²2. y=2x²3. y=x²4. y= ½x²
5. y=-5x²6. y=-x²7. y= -⅓x² Gets fatter and
opens up
Gets fatter and opens down
• Not every parabola has its vertex at the origin. The vertex can shift left, right, up, or down.
• The formula used when the vertex is NOT (0,0) is called vertex form:
y=a(x-h)²+k where (h,k) is the vertex
• Note: “h” shifts a parabola left and right “k” shifts a parabola up and down
(h, k)(x, y)
What else have we discovered?
• If “a” is the parabola opens up• If “a” is the parabola opens down• If “a” is a the parabola will be skinnier
(like 2 or 3)
• If “a” is a the parabola will be fatter(like ½ or ¼)
POSITIVENegative
BIGGER NUMBER
Smaller Number
Given a parabola with its vertex at (0,0) and a point on the parabola, write an equation and tell if the graph
opens up or down.
• A) Point- (1,2)y=ax²
2=a(1)²2=ax1
2=ay=2x²
Opens up because“a” is positive
• Label point (x,y)• Write formula• Plug in x and y• Solve for “a”
• Rewrite equation using “a”
(x and y stay the same)
(x,y)• B) Point- (-1,6)
y=ax²6=a(-1)²
6=1a6=a
Y=6x²Opens up because
“a” is positive
(x,y)
Write the equation of each parabola in Form
Vertex (h,k): (0,-4) Point on Graph (x,y): (2,0)
Plug in (h,k) and (x,y)and solve for “a”
• y=a(x-h)²+k• 0=a(2-0)²-4• 0=a(2)²-4• 0=4a-4• 4=4a• 1=a
Write equation of parabola by plugging in “a” and (h,k)
• y=a(x-h)²+k• y=1(x-0)²-4• y=1x²-4 or y=x²-4
Vertex y=a(x-h)²+k
*see graphOn your paper*
Try the next graph…
• Vertex: (2,4)• Point on graph: (1,1)
• Plug in h, k, x, and y, and then solve for “a”.
• y=a(x-h)²+k• 1=a(1-2)²+4• 1=a(-1)²+4• 1=1a+4• -3=1a• -3=a
• Write the equation of the parabola:
• y=a(x-h)²+k• y=3(x-2)²+4
Sketch the graph of each parabola. Label the vertex and axis of symmetry.
1. y=-½(x-2)²+3
Vertex: (2,3)Opens (Negative)x y y=-½(0-2)²+3 0 1 -½(-2)²+3 -½(4)+3 -2+3=1
Try the next graph…
2. y=3(x+2)²+4
Vertex: (-2,4)Opens (Positive)x y y=3(0+2)²+4 0 16 3(2)²+4 3(4)+4 12+4=1