Graphing Linear

Equations

Linear Equation

An equation for which the graph

is a line

SolutionAny ordered pair of numbers that makes a linear equation true.

(9,0) IS ONE SOLUTIONFOR Y = X - 9

Linear Equation

Example:

y = x + 3

Graphing

Step 1: ~ Three Point Method

~

Choose 3 values for x

GraphingStep 2:

Find solutions using tabley = x + 3

Y | X 0 1 2

Graphing

Step 3:Graph the points

from the table(0,3) (1,4) (2,5)

Graphing

Step 4:Draw a line to connect them

Try These

Graph using a table (3 point method)

1) y = x + 3

2) y = x - 4

X-intercept

Where the line crosses the x-

axis

X-intercept

The x-intercept has a y

coordinate of ZERO

X-intercept

To find the x-intercept, plug in ZERO for y

and solve

Slope

Describes the steepness of

a line

Slope

Equal to:

Rise Run

Rise

The change vertically, the change in y

Run

The change horizontally

or the change in x

Finding Slope

Step 1:Find 2 points on a

line(2, 3) (5, 4)(x1, y1) (x2, y2)

Finding Slope

Step 2:Find the RISE

between these 2 points

Y2 - Y1 =

4 - 3 = 1

Finding Slope

Step 3:Find the RUN between

these 2 points

X2 - X1 =

5 - 2 = 3

Finding SlopeStep 4:

Write the RISE over RUN as a ratio

Y2 - Y1 = 1

X2 - X1 3

Y-intercept

Where the line crosses the y-

axis

Y-intercept

The y-intercept has an x-

coordinate of ZERO

Y-intercept

To find the y-intercept, plug in ZERO for x

and solve

Slope-Intercept

y = mx + b

m = slopeb = y-intercept

Step 1:

Mark a point on the y-intercept

Step 2:

Define slope as a

fraction...

Step 3: Numerator is the vertical change

(RISE)

Step 4:

Denominator is the

horizontal change(RUN)

Step 5:

Graph at least 3 points and connect the

dots

Graphing Quadratic Functions

Definitions 3 forms for a quad. function

Steps for graphing each form Examples

Changing between eqn. forms

Quadratic Function A function of the form

y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola.

Example quadratic equation:

Vertex-

The lowest or highest pointof a parabola.

Vertex

Axis of symmetry-

The vertical line through the vertex of the parabola.

Axis ofSymmetry

Standard Form Equationy=ax2 + bx + c

If a is positive, u opens upIf a is negative, u opens down

The x-coordinate of the vertex is at To find the y-coordinate of the vertex, plug the

x-coordinate into the given eqn. The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex x-

coordinate. Use the eqn to find the corresponding y-values.

Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

a

b

2

Example 1: Graph y=2x2-8x+6

a=2 Since a is positive the parabola will open up.

Vertex: use b=-8 and a=2

Vertex is: (2,-2)

a

bx

2

24

8

)2(2

)8(

x

26168

6)2(8)2(2 2

y

y

• Axis of symmetry is the vertical line x=2

• Table of values for other points: x y

0 6 1 0 2 -2 3 0 4 6

* Graph!x=2

Now you try one!

y=-x2+x+12

* Open up or down?* Vertex?

* Axis of symmetry?* Table of values with 5

points?

(-1,10)

(-2,6)

(2,10)

(3,6)

X = .5

(.5,12)

Vertex Form Equationy=a(x-h)2+k

If a is positive, parabola opens upIf a is negative, parabola opens down.

The vertex is the point (h,k). The axis of symmetry is the vertical

line x=h. Don’t forget about 2 points on either

side of the vertex! (5 points total!)

Now you try one!

y=2(x-1)2+3

Open up or down? Vertex?

Axis of symmetry? Table of values with 5 points?

Example 2: Graphy=-.5(x+3)2+4

a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y

-1 2 -2 3.5

-3 4 -4 3.5 -5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

Intercept Form Equation

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

The x-intercepts are the points (p,0) and (q,0).

The axis of symmetry is the vertical line x= The x-coordinate of the vertex is To find the y-coordinate of the vertex, plug

the x-coord. into the equation and solve for y.

If a is positive, parabola opens upIf a is negative, parabola opens down.

2

qp 2

qp

Example 3: Graph y=-(x+2)(x-4)

Since a is negative, parabola opens down.

The x-intercepts are (-2,0) and (4,0)

To find the x-coord. of the vertex, use

To find the y-coord., plug 1 in for x.

Vertex (1,9)

2

qp

12

2

2

42

x

9)3)(3()41)(21( y

• The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)

x=1

(-2,0) (4,0)

(1,9)

Now you try one!

y=2(x-3)(x+1)

Open up or down? X-intercepts?

Vertex? Axis of symmetry?

(-1,0) (3,0)

(1,-8)

x=1

Changing from vertex or intercepts form to standard form

The key is to FOIL! (first, outside, inside, last)

Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8 =-(x2-9x+4x-36) =3(x-1)(x-

1)+8 =-(x2-5x-36) =3(x2-x-

x+1)+8y=-x2+5x+36 =3(x2-

2x+1)+8 =3x2-

6x+3+8 y=3x2-6x+11