Properties of Parabolas

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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