Math 31 - Ch 3. - Graphing Linear Equations in Two Variables
Using Intercepts to Graph Ax + By = C
1. Find the x-intercept. Let y = 0 and solve for x. 2. Find the y-intercept. Let x = 0 and solve for y. 3. Find a checkpoint, a third ordered-pair solution. 4. Graph the equation by drawing a line through the three points.
Slope and Parallel Lines
1. If two nonvertical lines are parallel, then they have the same slope. 2. If two distinct nonvertical lines have the same slope, then they are parallel. 3. Two distinct vertical lines, each with undefined slope, are parallel.
Slope and Perpendicular Lines
1. If two nonvertical lines are perpendicular, then the product of their slopes is -1. 2. If the product of the slopes of two lines is -1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
How to Graph an Inequality in Two Variables
1. Replace the inequality symbol with an equal sign .... 2. Graph the corresponding linear requisition. (Using method for Standard Form.)) 3.1 Draw a solid line ____ IF original inequality contains a </ or \> ect symbol. 3.2 Draw a dashed line - - - IF the original inequality contains a < or > symbol. 4.1 Choose a test point from one of the half planes. (Do not choose a point IN the line) 4.2 Sub the coordinates of the test point into the inequality 5.1 If TRUE statement results - shade half the plane WITH the test point. 5.2 If false statement, shade half plane without test point. NOTE - Test points are only necessary with *Standard Form*.
Graphing Linear Inequalities Without Test Points
If using y>mx+b or y<mx+b you don't need a test point. - If y>mx+b = Shade above the line - If y<mx+b = Shade below the line
Plotting/Plot
Locating points corresponding with Ordered Pairs.
Lines M in Linear Equations
M determines the line's steepness. Any real numbers can be used for M.
Ax + By = 0
Passes through the origin. Can be graphed using the origin as one point on the line. - Find two other points by finding two other solutions of the equation. - Select values for either variable, other than 0, and find corresponding values for the other variable.
Graph of the Equation
Set of all points whose coordinates satisfy the equation.
Slope
The steepness of a line on a graph
Rise
The vertical change between any two points on a line
X-Intercept
The x-coordinate of a point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is *always zero.*
m Variable
This is the slope of the line represented.
Finding Ordered Pairs that are Solutions of an Equation in Two Variables
- Select a value for one of the variables - Substitute that value into the equation and find the corresponding value of the other variable - Use the values of the two variables to form an ordered pair (x, y). This pair of a solution of the equation.
Graphing Ax + By = C by Using the Slope and y-Intercept
1. Begin by solving Ax + By = C for y. This will put the equation in slope-intercept form. 2. Then use the three-step procedure to graph the equation.
Point Plotting Method
1. Find several ordered pairs that are solutions of the equation. 2. Plot these ordered pairs as points in the Rectangular Coordinate System. 3. Connect the Points with a smooth curve out line.
Intercepts of a line
A point where a line intersects a coordinate axis is called an intercept.
Y- Intercept
The y-coordinate of a point where a graph crosses the y-axis. *The x-coordinate corresponding to a y-intercept is always zero.*
Graphing Horizontal or Vertical Inequalities
These do not require test points, either. Vertical Lines: a = x - If x>a - shade to right of x=a - If x<a - shade to the left of x=a Horizontal Lines: b = y - If y>b shade above the line - If y<b shade below line
Parallel
Two non intersecting lines that lie in the same plane. The have the same slope. The same "steepness."
y-axis
Vertical axis on a coordinate plane. Positive numbers above. Negative numbers below.
Point-Slope Form of the Equation of a (nonvertical) Line
use only if: 1. we know slope 2. and point not containing y-intercept y - y1 = m(x - x1) you will never substitute numbers for x and y. only for x1 and y1 and m.
Slope-Intercept Form of the Equation of a (non vertical) Line
y = mx + b use with slope and y intercept
Standard Form
Ax+By=C
Ordered Pair of Real Numbers
Each point in the Rectangular Coordinate System corresponds to an ordered Pair. (0, 0)
Linear Equations in Two Variables
Equations like y = 3x and y = 2x
X-Coordinate
First number of the Ordered Pair. Horizontal locations.
Quadrants of the Rectangular Coordinate System
Four regions of the Rectangular Coordinate System which divide into I, II, III, IV.
x-axis
Horizontal number line in a coordinate plane. Positive numbers to the right. Negative numbers to the left.
Ax + By = C
All equations in the form of Ax + By = C are straight lines when graphed as long as A and B are not both zero.
Solution of an Inequality in Two Variables
An ordered pair or ordered pairs that make the inequality true.p
Negative Reciprocals
Reciprocals with opposite signs. Two nonvertical line are perpendicular if the slope of one is the negative reciprocal of the slope of the other.
Scatter Plot
Set of points representing data.
Graph of Inequality in Two Variables
Similar to graphing an equation in two variables.
Calculating Slope
Slope = Rise/Run
Run
The horizontal change between any two points on a line
Origin
The point of intersection of the axes. X and y axes zero points.
Y-Coordinate
The second number in an Ordered Pair. Vertical locations.
Linear Inequality in Two Variables
ax + by > c; ax + by < c; y > mx + b; y < mx + b
Lines b in Linear Equations
b determines the pint where the line crosses he y-axis. Any real numbers can be used for b.
Equations of Horizontal and Vertical Lines
If A or B in the equation Ax + By = C turns out to be zero you end up with vertical lines.
Graphing y = mx + b by Using the Slope and y-Intercept
If a line's equation is written with y isolated on one side, we can use the y-intercept and the slope to obtain its graph. 1. Plot the point containing the y-intercept on the y-axis. This is the point (0,b). 2. Obtain a second point using the slope, m. Write m as a fraction, and use the rise over run, starting at the point on the y-axis, to plot this point. 3. Draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.
Perpendicular
Two lines that intersect at a right angle, 90 degrees.
Checkpoint
When graphing using intercepts it's always a good idea to use a third point *checkpoint* before drawing a line. 1. Select a value for either variable, other than 0. 2. Find the corresponding value for the other variable. Checkpoint should lie on the same line as the x and y intercepts. Recheck your work if it does not.
A Solution of an Equation in Two Variables (x and y)
When the x-coordinate and y-coordinate are substituted for x and y in the equation, we get a true statement.