UNIT 16 CONIC SECTIONS DAY 4: PARABOLA

1 APPLICATION OF PARABOLAS

Why are parabolas used to build bridges?

2 WHAT DOES A PARABOLA REMIND YOU OF??

What is the graph to the right?
How would you describe this?
What if there was a “-” in front of x?

3 PARABOLA VOCABULARY

Center/vertex : the point (h, k) where the parabola makes it’s sharpest turn
directrix : a line from which distances are measured in forming a conic
focus : a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus”
axis of symmetry: the line that splits the parabola up the middle
end of focal cord (EOFC) : a line that is parallel to the directrix and passes through the focus 4 PARABOLA

A parabola is the locus of points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the fixed line.

d1 = d 2 2p
p is the distance from the directrix

to the vertex d1

p is the distance from the vertex p d2 to the focus p 5 PARABOLA EQUATIONS

Standard form: − ℎ = − axis of symmetry is vertical − = − ℎ axis of symmetry is horizontal

= 1

6 GRAPHING PARABOLAS

Graph + 1 = 8 + 1
Vertex: (-1,-1)
= 8
= 2
Focus: (1,-1)
Directrix: x = -3
EOFC: (1, -5) and (1,3)

7 YOU TRY!

Graph − 1 + 8 + 2 = 0
Vertex: (1,-2)
= 8
= 2 Opens Down
Focus: (1,-4)
Directrix: y = 0
EOFC: (-3, -4) and (5,-4)

8 WRITING EQUATIONS FROM A GRAPH

Write the equation of the parabola shown in the graph
From the graph
2p = 3
The vertex is at (0,-2)
− ℎ = −
= 6 + 2

9 YOU TRY!

Write the equation of the hyperbola shown in the graph
From the graph
The vertex is at (3, 2)
p = 1
− = − ℎ
− 2 = − 3

10 WRITING EQUATIONS GIVEN CHARACTERISTICS

Write the equation of the parabola with a directrix at y = 6 and a focus at (0,-6).
The directrix is parallel to the x-axis
The line of symmetry is parallel to the y-axis (x = 0)
Parabola will open DOWN because the focus is below the directrix
2p = 6-(-6)=12, p = 6, and 4p = 24
Vertex is at (0,0)
− ℎ = − = −2 11 YOU TRY!

Write the equation of the parabola with focus at (2,4) and a vertex at (2,1).
Line of symmetry is parallel to the y-axis
Focus is above the vertex, parabola open UP
p = 4-1 = 3 and 4p = 12
− ℎ = −
− 2 = 12 − 1

12 CONVERTING BETWEEN FORMS

Find the standard form of the hyperbola from the general form
− − − + = 0
− − = − (Group terms)
− ( + 2) = − (Factor)
(+2 + 1) = −2 + 1 + 1 (Divide by -1/2 and Complete the square)
+ 1 = −2( − 1) 13 COMPARING EQUATIONS

Conic Section Standard Form General Form Parabola − ℎ = − − − − + = 0 Hyperbola − = 1 2 − + + 8 − 3 = 0 Ellipse + = 1 + − 8 + − 8 = 0 Circle + − 3 = 8 2 + 2 −8+12+2=0

**Parabolas only have 1 squared varable! 14 HOW TO CLASSIFY WHEN THE XY TERM IS INCLUDED

The xy term will tilt the graph.
The major(minor) axis/transverse axis/axis of symmetry will not be parallel to the x or y axes

To classify the conic section, you need to use the discriminant −
Ellipse: − 0 Hyperbola: − 0
Circle: − 0 = Parabola: − = 0 = 0 15 EXIT TICKET

What is importance of the focus of a parabola?

WebAssign due on Monday, April 10.

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