Parabola

Quadratic Equations and Parabola

ParabolaA parabola is a curve where any point is at an equal distance from a fixed point (the focus and a fixed straight line (the directrix).

Source: http://www.mathsisfun.com/geometry/parabola.html

• the directrix and focus

• the axis of symmetry (goes through the focus, at right angles to the directrix)

• the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

Vocabulary

Source: http://www.mathsisfun.com/geometry/parabola.html

EquationsStandard form:

Vertex form

GraphingSteps:a. Identify the orientation

If a is positive, then the parabola opens upward.If a is negative, then the parabola opens downward.Example: opens upward opens downward

GraphingSteps:b. Find the vertex

1. , vertex:

2. , vertex:

3. , vertex:

4. , vertex:

GraphingSteps:c. Find the -intercept

d. Find the -intercept

e. Find other coordinates

f. Find the axis of symmetry

g. Plot the points

Graphing

a. Since is positive, the parabola opens upward.b. vertex: c. -intercept

set

-intercept:

1. 𝑦=𝑎𝑥2

Example: 𝑦=𝑥2

Graphing1. 𝑦=𝑎𝑥2

Example: 𝑦=𝑥2

d. -interceptset

-intercept: e. Other coordinates


Example: 𝑦=𝑥2

f. axis of symmetry:

g. plot the points


Example: 𝑦=𝑥2

(0,0) (1,1)(-1,1)

(2,4)(-2,4)

(3,9)(-3,9)

Graphing

a. Since 2 is negative, the parabola opens downward.b. vertex: c. -intercept

set

-intercept:

1. 𝑦=𝑎𝑥2

Example: 𝑦=−2𝑥2



d. -interceptset




f. axis of symmetry:

g. plot the points


Example: 𝑦=−2𝑥2(0,0)(1,-2)

(-3,-18)

(2,-8)(-2,-8)

(3,-18)

(-3,-2)

Graphing

a. Since 2 is positive, the parabola opens upward.b. vertex: c. -intercept

set -intercept: none

1. 𝑦=𝑎𝑥2+𝑘Example: 𝑦=2 𝑥2+1

Graphing+k

Example: 𝑦=2 𝑥2+1d. -intercept

set


Graphing1. 𝑦=𝑎𝑥2+𝑘Example: 𝑦=2 𝑥2+1f. axis of symmetry:

g. plot the points

Graphing1. 𝑦=𝑎𝑥2+𝑘Example: 𝑦=2 𝑥2+1

(0, 1)

(2, 9)(-2, 9)

(1, 3)(-1, 3)

Education

Parabola