course: adv. alg. & trig. aim: graphing parabola do now: aim: how do we graph a parabola?

Course: Adv. Alg. & Trig.Aim: Graphing Parabola

Do Now:

Aim: How do we graph a parabola?

2 LV solve for A

KA


Graphing the parabola y = x2

Table of Values

Graph y = x2 for the values -3 < x < 3

(-3)2 9 -3,9

(-2)2 4 -2,4

(-1)2 1 -1,1

(0)2 0 0,0

(1)2 1 1,1

(2)2 4 2,4

(3)2 9 3,9

-1,1

(0,0)

(-2,4)

(-3,9)

(1,1)

(2,4)

(3,9)

y =

x2

Axis of symmetry-Turning point-

x-intercept & y-intercept -

x = 0(0,0)

Minimum

(0,0)

x

y


Graphing the parabola y = x2 – 4

Table of Values

Graph y = x2 – 4 for the values -3 < x < 3

(-3)2- 4 5 -3,5

(-2)2- 4 0 -2,0

(-1)2- 4 -3 -1,-3

(0)2- 4 -4 0,-4

(1)2- 4 -3 1,-3

(2)2- 4 0 2,0

(3)2- 4 5 3,5

-1,-3

(0,-4)

(-2,0)

(-3,5)

(1,-3)

(2,0)

(3,5)

y =

x2 -

4


x-intercepts-

(0, -4)x = 0

Minimum

(-2, 0)&(2, 0)


Graph y = -x2 +2x + 5 for the values -2 < x < 4

Table of Values

-(-2)2+2(-2)+5 -3 -2,-3

(-1,2)

(0,5)

(-2,-3)

(4,-3)

(3,2)

(2,5)

(1,6)


x-intercepts-

(1,6)x = 1

Maximum

-(-1)2+2(-1)+5 2 -1,2

-(0)2+2(0)+5 5 0,5

-(1)2+2(1)+5 6 1,6

-(2)2+2(2)+5 5 2,5

-(3)2+2(3)+5 2 3,2

-(4)2+2(4)+5 -3 4,-3

(-?,0)&(?,0)

y =

-x2 +

2x +

5

How could we find the values of the x-intercepts or roots of y = -x2 + 2x + 5?


A parabola is symmetrical about a line called the

axis of symmetry.

y = ax2 + bx + c

axis

of

sym

met

ry

turning point


Axis of symmetry of a parabola y = ax2 + bx + c is the equation

The x-value of the turning point equals -b/2a and the y-value can be found by

substituting -b/2a for x in the equation y = ax2 + bx + c.

x b

2a

x b

2a

Axis of symmetry

Axis of symmetry and the turning point

Turning point of the parabola is always found on the axis of symmetry.


When a > 0 the parabola is a “valley” that

opens upward.

The curve falling until it reaches a lowest point, a

minimum point. Then the curve

turns and begins to rise (turning

point).

When a < 0 the parabola is a

“hill” that opens downward.

The curve is rising until it

reaches a highest point, a maximum point. Then the curve turns and

begins to fall (turning point).

y = ax2 + bx + c


The absolute value of a determines “fatness”.

y = ax2 + bx + c

a = 1

a < 1

a > 1

As a increases, the shape of the parabola gets “thinner”.

As a decreases in value, the shape of the parabola gets fatter or wider.


The value of c is the y-intercept of the parabola or the point where the parabola crosses the y-axis.

y = ax2 + bx + c

y = ax2 + bx + c

y = ax2 + bx + c

y = ax2 + bx + c

y = ax2 + bx + c

c = +10

c = +5

c = 0

c = -6

Ex. y = 4.34x2 - 15.445x + 1.456

y-intercept is +1.456


Model Problem

Write an equation of the axis of symmetry of the graph of y = 3x2 + 12x – 2 and find the coordinates of the turning point.

1) Establish what a, b and c are for the equation y = 3x2 + 12x – 2

x b

2aThe equation for finding the axis of symmetry.

2) Evaluate for x = -b/2a

a = 3 b = 12 c = -2

x = -12/2(3) = -12/6 = -2

3) To find turning point, evaluate y = 3x2 + 12x – 2 when x = -2

Axis of symmetry is equation x = -2

y = 3(-2)2 + 12(-2) – 2 = -14

coordinates of turning point - (-2, -14)


Model Problems

Which is an equation of the graph shown?

1) y = x2 – 4x + 4

2) y = x2 + 4x + 4

3) y = -x2 – 4x + 4

4) y = -x2 + 4x + 4

Since the parabola opens upward eliminate 3) and 4).

Find the axis of symmetry for equation 1) equation 2)

x = -b/2a = -(-4)/2(1) = 2x = -b/2a = -(4)/2(1) = -2

x =

-2

Since a. of s. of shown graph is x = -2, choice 2) is correct answer.


Model Problem – How to Start

Table of Values

Graph y = x2 - 4x + 3

(-1)2-4(-1)+3 8 -1,8

(0,3)

(-1,8)

(1,0) (2,-1)

(3,0)

(4,3)

(1,8)

x =

2

(0)2-4(0)+3 3 0,3

(1)2-4(1)+3 0 1,0

(2)2-4(2)+3 -1 2,-1

(3)2-4(3)+3 0 3,0

(4)2-4(4)+3 3 4,3

(5)2-4(5)+3 8 5,8

Since interval for table of values is not given, find the axis of symmetry and use 3 interval values to the left and 3 interval values to the right of the axis.

x b

2aAxis of symmetry - x = -(-4)/2(1) = 2

Values for table: -1, 0, 1, 2, 3, 4, 5


y = x2 and x = y2

What is the difference between y = x2 and

x = y2?

y =

x2

Describe the transformation that changes y = x2 to x = y2.

x = y2

y x

Rotation of 900 about the origin.


Do Now:

Aim: How do we graph a parabola?

Write an equation of the axis of symmetry of the graph of y = 3x2 + 12x – 2 and find the coordinates of the turning point.

course: adv. alg. & trig. aim: graphing parabola do now: aim: how do we graph a parabola?

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