Chapter 9 Linear equations/graphing 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane
Rectangular Coordinate System Quadrant II (-,+) y-axis Quadrant I (+,+) x-axis Origin (0,0) Quadrant III (-,-) Quadrant IV (+,-)
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; 1 st degree)
Solution Solution: 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.
Slope To find the slope: Choose 2 points on the line: point 1: (x 1, y 1 ) point 2: (x 2, y 2 ) Slope = y 2 - y 1 = change in y = rise x 2 - x 1 change in x run
Slope The letter m is often used to represent the slope: m = rise = y 2 - y 1 run x 2 - x 1 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 y 1 = m(x x 1 ) (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 Graphing 1. 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 2 nd 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 = y 2 - y 1 x 2 - x 1 Plug m and b into the slope intercept equation: y= mx + b
Point Slope Form of a Line If the y-intercept is not given, use the point slope form to find the equation of the line. y y = m( x x1) 1 m is the slope, (x 1, y 1 ) is a point on the line If given 2 points: 1. Find the slope and plug it in for m: m 2. Plug in one of the points for (x 1, y 1 ). = ( y ( x 2 2 y x 1 1 ) ) If given the slope and 1 point: plug in the slope and the point into the point slope equation.
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)
Evaluating a Function 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