Graphing Linear Equations PPT.ppt

Topic: Graphing

Linear Equations

Objectives

Understand the difference between linear and nonlinear equations Find solutions to linear equations

Know how to calculate slope given a graph or coordinates. Understand how slope how to interpret the slope and how it relates to graphs.

Write linear equations in various forms. Learn the formula for each.

Graph linear equations.

Know which method to use for graphing based on the given information.

Standards

Linear equation – An equation whose solutions form a straight line on a coordinate plane.

Solutions of a Linear Equation – any point that lies on the line (or) any ordered pair that is true for the equation.

Slope – A measure of the steepness of a line on a graph; rise divided by the run.

x-intercept – the x-coordinate of the point where the graph of a line crosses the x-axis (where y = 0).

y-intercept – the y-coordinate of the point where the graph of

a line crosses the y-axis (where x = 0).

Slope-intercept form (of an equation) – a linear equation written in the form y = mx +b, where m represents slope and

b represents the y-intercept.

Part 1

What Are Linear Equations?

How do you find solutions to

Linear Vs. Nonlinear

A linear equation is an equation whose solutions are graphed with a straight line.

Linear – makes a straight line Linear – makes a

straight line

Non-Linear Not a straight line

Solutions to Linear Equations

Given an ordered pair

– Substitute the x and y values into the equation.

– If the left side = right side, the ordered pair is a

solution

Given a graph

Checking An Ordered Pair

1. Substitute the x and y values into the equation.

– Remember in an ordered pair the x always comes before the y (x,y)

2. Simplify

3. If the left = right, the ordered pair is a solution

If the left ≠ right, the ordered pair is not a solution.

Solution

Solution Not a

All solutions of a

particular linear equation fall on the line, and all the points on the line are solutions of the equation. Look at the graph to the left, points (1, 3) and (3, -5) are found on the line and are solutions to the equation.

Part 2

Methods of Graphing Linear

Equations

1. Input/Output Tables

2. Slope & One Coordinate

3. Direct Variation

4. Slope Intercept

5. X & Y Intercepts

6. Point Slope

Various Forms of an Equation of a

Line.

Slope

Direct

Variation

Various Forms of an Equation of a

Line.

Slope-Intercept Form

Standard Form

Point-Slope Form

slope of the line

intercept

y mx b

m

b y







 

, , and are integers

0, must be postive

Ax By C

A B

C

A

A















slope of the line

,

is any point

y y

m x x

m

x y









Graph Using Input-Output

Tables

Input/Output Tables

1. Pick some values to substitute into the

equation for x.

– Select at least 3 values (one positive, zero, and

one negative)

2. Substitute these values and solve for y.

3. These values become your ordered pair

(x,y).

Complete the table below, then

graph and tell whether it is linear.

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

-1 0

Can you determine if the equation is linear?

The equation y = 2x + 3 is a linear equation

because it is the graph of a straight line. Each

time

x increases

by 1 unit,

y increases

by 2.

X y = 2x + 3 Y (x,y) -2 2 (-2) + 3 -1 (-2, 1)

The equation y = 2x + 6

is a linear equation

because it is the graph of a straight line and each time x increases by 1 unit, y increases by 2

X Y = 2x + 6 Y (x, y)

1 2(1) + 6 8 (1, 8)

2 2(2) + 6 10 (2, 10)

3 2(3) + 6 12 (3, 12)

4 2(4) + 6 14 (4, 14)

Real world example

The graph (c = 5x + 10)

at the left shows the cost for Company A cell

phone charges.

Slope

Slope of a line is its rate of change. The following example describes how slope (rate of change) is applied.

Rate of change is also know as steepness or pitch, or rise over run. Change is often symbolized in

Slopes: positive, negative, no slope (zero), undefined. Positive (uphill) Positive (uphill) Negative (downhill) Negative (downhill) Zero (flat)

Zero (flat) Undefined (not possible) Undefined (not

Finding slope (rate of change) using a graph and two points.

1. Connect the 2 points by using the gridlines to make a right triangle 2. Count the rise

3. Count the run

If an equation is linear, a constant change in the x-value

corresponds to a constant change in the y-value. The graph shows an example where each time the x-value increases by 3, the y-value increases by 2.

2

2 3

3

Given the point (4, 2), find the slope of this line?

To make finding slope easier, find

where the line crosses at an x

Find the slope of the line that passes through each pair of points.

1) (1, 3) and (2, 4)

2) (0, 0) and (6, -3)

3) (2, -5) and (1, -2)

4) (3, 1) and (0, 3)

Finding Slope

Use the graph of the line to determine its slope. Choose two points on the line (-4, 4) and (8, -2). Count the rise over run or you can use the slope

formula. Notice if you switch (x₁, y₁) and (x₂, y₂), you get the same slope: 4 - (-2) = 6 or (-2) – 4 = -6

Graphing a Line Using a Point and the Slope

Graph the line passing

through (1, 3) with slope 2 1. Graph the ordered pair.

2. Count the slope to graph the 2nd point (rise/run)

Direct Variation

y=kx

Steps to Graphing with Direct

Variation

1. Graph the 1

point at the origin

2. Identify the slope from the equation (y=kx,

k is the slope)

3. Count the slope to graph the 2

point.

Slope Intercept

y = mx + b

Slope-intercept

Form

An equation whose

graph is a straight

line is a

linear

equation

. Since a

function rule is an

equation, a

function can also

be linear.

m = slope

b = y-intercept

Y =

m

x +

b

(if you know the slope and where the line crosses the y-axis, use this

Graph Using Slope Intercept

1. Find the slope and the y-intercept

(m=slope, b = y intercept)

2. Plot the y-intercept

3. Count the slope (rise/run) to

plot the 2

point.

X & Y Intercepts

Use when the equation is in

Use when the equation is in

Standard Form of

an Equation

The standard form of a linear equation, you can use the x- and

y-intercepts to make a graph.

The x-intercept is the x-value of the point where the line crosses.

The y-intercept is the y-value of the point where the line crosses.

Ax + By = C

To graph a linear equation in standard form, you find the x-intercept by substituting 0 for y and solving for x. Then

substitute 0 for x and solve for y.

2x + 3y = 6

2x + 3(

0

) = 6

2x = 6

x = 3

The x-intercept is 3. (y = 0)

2x + 3y = 6

2(

0

) + 3y = 6

3y = 6

y = 2

Let’s take a look at that equation again!

2x + 3y = 6 2x + 3y = 6

2x = 6 3y = 6 x = 3 y = 2

Since you are substituting (0) in for one variable and solving for the other, any number multiplied times (0) = 0.

So, in the first example 3(0) = 0, and in the second example 2(0) = 0.

Since 3(0) = 0, just cover up the 3y and

solve what’s

left.

Again, since 2(0) = 0, just

Find the x-intercept and y-intercept of each line. Use the intercepts to graph the equation.

1) x – y = 5

2) 2x + 3y = 12

3) 4x = 12 + 3y

4) 2x + y = 7

Point Slope

Use when you know the slope and

Use when you know the slope and

one coordinate

one coordinate

Point-Slope Form

and Writing

Equations

Suppose you know that a line passes through the point (3, 4) with slope 2. You can quickly write an equation of the line using the x- and y-coordinates of the point and using the

slope.

The point-slope form of the equation of a nonvertical line that passes through the (x₁, y₁) with slope m.

y –

y

=

m

(x –

x

)