Properties of Parabolas We Parabolas puh-rab-uh-luh

Axis of symmetry 1. Vertex- the highest or lowest point on a parabola 2. 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

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

See what happens when we put in the following numbers for “a”. Use your graphing calculator 1.y=5x² 2.y=2x² 3.y=x² 4.y= ½x² 5.y=-5x² 6.y=-x² 7.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” isthe 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)

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)² 2=ax1 2=a y=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=ax² 6=a(-1)² 6=1a 6=a Y=6x² Opens 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 *see graph On 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 ½(-2)²+3 -½(4) =1

Try the next graph… 2. y=3(x+2)²+4 Vertex: (-2,4) Opens (Positive) x y y=3(0+2)² (2)²+4 3(4) =1