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C Graphing Quadratic FunctionsThe graph of a quadratic function is a U-shaped graph called a parabola. When yougraph a quadratic function you find ordered pairs that satisfy the function.

Rules for Graphing a Quadratic Function1. Find the coordinates of the vertex.

2. Create an input/output table.3. Select several other values for x.

Substitute the values for x into the equation. Solve for y.4. Plot each ordered pair on the coordinate plane. Draw a parabola.

ExampleGraph. y = -2x2 + 1Step 1 Find the coordinates of the vertex. din f -b -0 0

x-coor ate 0 vertex = 2a = 2(-2) =Y = - 2xl + 1 = -2 (0) 2 +l = 1

The vertex is at (0, 1).

Step 2 Create an input/output table.

Step 3 Select several other values for x.Substitute the values for x into the

equation. Solve for y.

X-2x2 + 1

y2

-2(2)2 + 1-7-2

-2(-2)2 + 1-71

-2(1)2 + 1-1-1

-2(-1)2+1-1

Step 4 Plot each ordered pair on thecoordinate plane. Draw a parabolaconnecting the points.

Practice AGraph each function.

1. y = xl + 2x + 3Find the coordinates of the vertex. din f b-

x-coor ate 0 vertex = - 2a = 2() = --

y= __ 2+2( __ ) +3= __

The vertex is at __ .

Xx2 + 2x+ 3

y-2

(-2)2 + 2(-2) + 332

__ 2 + 2(__ ) + 3--a

__ 2 + 2(__ ) + 3--1

__ 2 + 2(__ ) + 3--Select several other values for x.Substitute the values for x into the

equation. Solve for y.

Plot each ordered pair on thecoordinate plane. Draw a parabolaconnecting the points.

2. y = xl + 2 _

3. y = 3xl - 3x - 1 _

Create an input/output table.

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C Properties of a Graph of a Quadratic FunctionThe graph of a quadratic function is a U-shaped graph called a

parabola. The vertex of a parabola is the lowest point of a parabolathat opens up. In a parabola that opens down, the vertex is the

highest point of the parabola. The line passing through that divides

the parabola into two equal parts is the axis of symmetry.

opens "up"

Properties of the Graph of a ParabolaFor quadratic equations in the form y = ax?- + bx + c:1. If a is positive, the parabola opens up; if a is negative, the parabola opens down.

2. To find the x-coordinate of the vertex, use ;:; plug the x-value into the equationto find the y-coordinate of the vertex.

3. To find the axis of symmetry, use x = ;:.

ExampleFor the quadratic equation y = 2x2 - 4x - 3, tell whether the parabola opens up ordown and find the coordinates of the vertex and the axis of symmetry.

Step 3 To find the axis of symmetry, use-bx= 2a.

Step 1

Step 2

If a is positive, the parabola opens up; a is positive, the parabola opens up.

if a is negative, the parabola opens down.

To find the x-coordinate of the vertex, The x-coordinate of the vertex: ;: = -;~/ = 1

use ;:; plug the x-value into the y = 2x? - 4x - 3 = 2(1)2 - 4(1) - 3 = -5equation to find the y-coordinate of "\T (1 5)vertex: ,­the vertex.

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

Practice "'B

For each quadratic function tell whether the parabola opens up or down and findthe coordinates of the vertex and the axis of symmetry.

1. y = -x? - 4x + 2

If a is positive, the parabola opens up;if a is negative, the parabola opens down.

To find the x-coordinate of the vertex,

use ;:; plug the x-value into theequation to find the y-coordinate ofthe vertex.

To find the axis of symmetry, use-bx= 2a"

2. y = 2x? _

3. y=-4x?+ 8x _

a IS ; the parabola

opens _

The x-coordinate of the vertex:

-b _ -( )2a -2()

y = _(__ )2 - 4(__ ) + 2 = __

Vertex: _

-b -( )x=2"a=-2(--) =--

4. y = 3x? - 6x + 1 _

5. y = -lOx? + 5x + 3 _

Algebra 2

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C Writing a Quadratic Function from Its GraphA quadratic function can be written in the form y = a(x - hf + k. This form is known asthe vertex form of a quadratic function.

Rules for Writing a Quadratic Function from Its Graph1. Identify the vertex. The x-coordinate of the vertex is h, the y-coordinate is k.

2. Identify one other point on the graph. The x-coordinate of the point is x inthe vertex form, the y-coordinate is y in the vertex form.

3. Plug the values from Step 1 and Step 2 into the vertex form of a quadraticfunction. Solve for a.

4. Write the function using the vertex for hand k from Step 1 and a from Step 3.

ExampleA parabola has a vertex at (3. -1). Another point on the graph is at (0. 8). Writethe equation of the parabola in vertex form.

Vertex: (3, -1); h = 3, k=-1

Other point: (0, 8)

x= O,y= 8

Vertex: (1, -3); h = __ , k = __

8 = a(O - 3)2 + (-1)1=a

Step 1 Identify the vertex. The x-coordinateof the vertex is h, the y-coordinate ofthe vertex is k.

Step 2 Identify one other point on the graph.The x-coordinate of the point is x inthe vertex form, the y-coordinate is yin the vertex form.

Step 3 Plug the values from Step 1 andStep 2 into the vertex form of aquadratic function. Solve for a.

Step 4 Write the function using the vertex for y = 1(x - 3)2 - 1 = (x - 3)2 - 1hand k from Step 1 and a from Step 3.

Practice CFor each parabola. write an equation in vertex form.

1. Vertex: (1, -3); other point: (3, -5)

Identify the vertex. The x-coordinateof the vertex is h, the y-coordinate ofthe vertex is k.

a= __

Other point: ( 3, -5 )

x= __ ,y= __

y= __ (X- __ )2 + __

__ = a(__ - __ )2 + __

Identify one other point on the graph.The x-coordinate of the point is x inthe vertex form, the y-coordinate is yin the vertex form.

Plug the values from Step 1 andStep 2 into the vertex form of aquadratic function. Solve for a.

Write the function using the vertex for h

and k from Step 1 and a from Step 3.

2. Vertex: (1, -6); other point: (3,0) _

3. Vertex: (0, -3); other point: (3,0) _

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C Quadratic Functions in Intercept Form

A quadratic function can be written in the form y = a(x - p)(x - q). This form is knownas the intercept form of a quadratic function.

Rules for Writing a Quadratic Function in Intercept Form1. Find one of the x-intercepts. The x-coordinate is p.2. Find the other x-intercept. The x-coordinate of this intercept is q.3. Identify the x- and y-coordinates of the vertex. These coordinates

are x and y in the intercept form equation.4. Plug the values from Steps 1,2, and 3 into the intercept form of a

quadratic equation. Solve for a.

5. Write the equation using the values from Steps 1,2, and 4.

x-intercept: (2,0); P = 2

x-intercept: (-2,0); q =-2

2 = a(O - 2)(0 - (-2»)1

-"2= a

y= a(x- p)(x- q)

y= -~(x- 2)(x- (-2)

ExampleA parabola has a vertex at (0, 2) and x-intercepts at (2, 0) and (-2, 0). Write theequation of the parabola in intercept form.

Step 1 Find one of the x-intercepts. Thex-coordinate is p.

Step 2 Find the other x-intercept. Thex-coordinate of this intercept is q.

Step 3 Identify the x- and y-coordinates of the Vertex: (0,2); x = 0, y = 2vertex. These coordinates are x and y inthe intercept form equation.

Step 4 Plug the values from Steps 1,2, and 3into the intercept form of a quadraticequation. Solve for a.

Step 5 Write the equation using the valuesfrom Steps 1,2, and 4.

Practice .J::>

Write the equation of each parabola in intercept form.1. Vertex: (-1, -4); intercepts: 0,0) and (-3,0)

Step 1 x-intercept: 0,0); p = __Step 2 x-intercept: (-3,0); q = __

Step 3 Vertex: (-1, -4); x = __ , y = __

Step 4 __ = a(__ - __ ) (__ - __ )

__ =a

Step 5 Y = a(x - p)(x - q)

y= __ (x- __ ) (x- __ )

2. Vertex: (-2,3); intercepts: (-5,0) and (1,0) _

3. Vertex: 0, -2); intercepts: (0,0) and (2,0) _

Algebra 2

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