Parabolas

by

Dr. Carol A. Marinas

Transformation Shifts

Tell me the following information about:

f(x) = (x – 4)2 – 3 What shape is the graph? What direction is it going? What is the vertex? Is it a high point or low point? What is the y-intercept?

Transformation Shifts

Tell me the following information about:

f(x) = (x – 4)2 – 3 What shape is the graph? parabola What direction is it going? up What is the vertex? (4, –3) Is it a high point or low point? Low point What is the y-intercept? (0, 13)

y =(x – 4)2 – 3

Standard Form

• y = (x – h)2 + k

• Vertex is (h, k)

• Line of Symmetry is x = h

Standard to General Form

• y =(x – 4)2 – 3

• y = (x2 – 8x + 16) – 3

• y = x2 – 8x + 13

General to Standard Form

• y = x2 – 8x + 13

• y – 13 = x2 – 8x

• y – 13 + 16 = x2 – 8x + 16

y + 3 = (x – 4)2

• y = (x – 4)2 – 3

Vertex is (4, – 3)

• Get ‘a’ equal to 1 by multiplying or dividing the equation. (done)

• Move constant to left side

• Complete the square

• Solve for y

General Form

• y = ax2 + bx + c

Vertex is ( –b/2a, f(–b/2a) )

y-intercept is (0, c)

Line of Symmetry is x = –b/2a• Example: y = x2 – 8x + 13 Vertex is (–(– 8)/2(1), f (8/2)) or (4, f(4)) or (4, –3)

y-intercept is (0, 13)

Line of Symmetry is x = 4

x-intercepts

• For x-intercepts, the y value is 0.

• y = ax2 + bx + c becomes0 = ax2 + bx + c which is a quadratic

equation that is solved by factoring or the quadratic formula.

x-intercepts usingQuadratic Formula

• 0 = ax2 + bx + c

• x =

• b2 – 4ac is the discriminant and is used to tell us how many x-intercepts exist.

a

acbb

2

42

Discriminant

if less than 0, no x-interceptsb2 – 4ac if it is 0, 1 x-intercept if greater than 0, 2 x-intercepts

Example: y = x2 – 8x + 13The discriminant is (–8)2 – 4(1)(13) or 64 – 52 or 12 Since 12 > 0, there are 2 x-intercepts.

Finding the x-intercepts

Ex: y = x2 – 8x + 13 The discriminant is 12. • To actually find the x-intercepts, let’s continue

using the Quadratic Formula. • x = =

• x = 8 ± √12 = 8 ± 2√3 = 4 ± 2 2 The x-intercepts are (4 – , 0 ) and (4 + , 0)

a

acbb

2

42 )1(2

)13)(1(48)8( 2

3

3 3

Final Graph of y = (x – 4)2 – 3 or y = x2 – 8x + 13

Review

• Standard Form y = (x – h)2 + k

• General Form y = ax2 + bx + c

Know how to find the following:

* Vertex * y-intercept(s)

* High/Low Point * x-intercept(s)

* Axis of Symmetry * Graph the parabola