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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
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