Chapter 9 Linear equations/graphing

1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane

x-axis

y-axisQuadrant I

(+,+)

Quadrant IV (+,-)

Quadrant III (-,-)

Quadrant II (-,+)

Origin (0,0)

Rectangular Coordinate System

Plotting a Point

Point • A location on the graph. • (x,y), ‘ordered pair’ describes the point.

To graph point (x,y):

• start at the origin (0,0)

• move x units along the x-axis number line • Positive x, move right; Negative x, move left

• move y units based on the y-axis number line • Positive y, move up; Negative y, move down

– Graph lines on coordinate plane by: • Making a table of values and plot points • Using the intercept method • Using y = mx + b and graphing with y-intercept and

slope

Lines

• A line is made up of points connected together.

• Every line on a graph has a corresponding equation.

• Standard Form of Linear Equations Ax + B = C (One variable) Ax + By = C (Two Variables)

(A, B, C are real numbers, A & B are not both 0; 1st degree)

SolutionSolution: a numbers that makes the

equation true when they replace the variables.

If have more that one equation in a “linear system”, the numbers that replace the variables must make ALL the equations true.

SECTION 9.2

Graph an Equation Using Points“Point Plotting Method”

Generic Method – works for all types of equations.

1. Choose a value for one variable (either x or y). 2. Plug that value into the equation and solve for the other

variable. Now have a point (x,y). 3. Repeat steps 1 & 2 for additional points. 4. Plot the points. 5. Connect the dots.

To keep the problem organized, create a table. Remember, may have “f(x)” instead of “y”, but they are the same. x-values that are not in the domain will show as “gaps” on the graph.

Intercept Points

x-intercept • where the graph crosses the x-axis • points on the x-axis have a y-coordinate of 0: (x,0) • to find the x-intercept, plug in y=0 and solve for x.

y-intercept • where the graph crosses the y-axis • points on the y-axis have an x-coordinate of 0:

(0,y) • To find the y-intercept, plug in x=0 and solve for y.

Solutions of a Line

• A solution of an equation makes the equation true.

• All points on the graph of a line are solutions of the equation.

•  

SECTION 9.3

Study guide…

Slope of a line

The steepness of a line.

For any two points on the line, the slope is: = rise run

Rise over run.

SlopeTo find the slope:

Choose 2 points on the line: point 1: (x1, y1) point 2: (x2, y2)

Slope = y2 - y1 = change in y = rise x2 - x1 change in x run

SlopeThe letter ‘m’ is often used to represent the

slope: m = rise = y2 - y1

run x2 - x1

‘m’ is for the French word ‘monter’ – to rise.

There are 4 possibilities for the slope: • m>0; positive slope; line rises from left to right • m<0; negative slope; line falls from left to right • m=0; zero slope; horizontal line • m is undefined (zero in denominator of

fraction); vertical line

Horizontal Lines

• Horizontal lines have the form y = b, where b is the y-intercept. • y is always the same number. x can be any

number. • Slope, m = 0

Vertical Lines

• Vertical lines have the form x = n, where n is the x-intercept. • x is always the same number. y can be any

number. • the slope is undefined

SECTION 9.4

Parallel Lines • lines that never intersect. • lines have the same slope. • Vertical lines are parallel.

Perpendicular Lines • lines intersect at right angles. • slopes that are opposites and reciprocals of each other.

• Horizontal and vertical lines are perpendicular.

Forms of a Line

Standard Form: Ax + By = C

Slope-Intercept Form: y = mx + b

Point-Slope Form: y – y1 = m(x – x1)

(x and y are the variables; all other letters will be numbers in the equation.)

Slope-Intercept Form of a Line

y=mx+b

• m is the slope

• b is the y-intercept (0,b).

Slope Intercept Graphing1. Solve the equation for y: y = mx + b. 2. Plot the y-intercept first: (0,b) 3. Then use the slope, m, to find a 2nd

point. – Write ‘m’ as a fraction: rise over run. – Start at ‘b’ and use the slope as directions

to the next point: rise (numerator) over run (denominator).

4. Connect the dots.

Find the Equation from the Graph:Slope Intercept Form

Use this form of a line if the graph has the y-intercept point shown:

1. Find the y-intercept, b. (where the line crosses the y axis) 2. Find the slope, m

- use the graph and find the slope (rise over run) Or - choose 2 points on the line and use the slope formula m = y2 - y1

x2 - x1

Plug m and b into the slope intercept equation: y= mx + b

Point Slope Form of a LineIf the y-intercept is not given, use the point slope form to find

the equation of the line.

m is the slope, (x1, y1) is a point on the line

If given 2 points: 1. Find the slope and plug it in for m: 2. Plug in one of the points for (x1, y1).

If given the slope and 1 point: plug in the slope and the point into the point slope equation.

)( 11 xxmyy −=−

)()(

12

12

xxyy

m−

−=

Perpendicular lines have slopes that are opposite signs and reciprocals of each other.

Function Equations

If an equation is a function, then y = f(x).

Example: y = x + 1 is the same as f(x) = x + 1

Note: f(x) is a special notation; does NOT mean f • x (f times x)

Evaluate: plug in a value for x and solve for f(x).

Example: f(x) = x + 2; find f(3) (Note: This is the same as: Evaluate y = x+2 when x = 3)

Plug in 3 for x: f(3) = 3 + 2 f(3) = 5 Find f(0): f(0) = 0 + 2 f(0) = 2

Evaluating a Function