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ALGEBRA 2:
2.1 Represent Relations and Functions
Goal  Represent relations and graph linear functions.
VOCABULARY
Relation A mapping, or pairing, of input values with output values
Domain The set of input values in a relation
Range The set of output values in a relation
Function A relation for which each input has exactly one output
Equation in two variables An equation that has an independent or input variable and a dependent or output
variable that depends on the value of the input variable
Linear function A function that can be written in the form y = mx+ b where m and b are constants
REPRESENTING RELATIONS
A relation can be represented in the following ways:
Ordered
Table
Graph
Mapping Diagram
Pairs
Input Output
(2, 2)
x
y
(2, 2)
2
2
2
2
(0, 1)
2
2
0
2
(3, 1)
3
1
0
1
3
1
Example 1
Identify domain and range.
Identify the domain and range of the given relation. Then represent the relation using a graph and a mapping
diagram.
(5,-2), (-3, -2), (3,3), (-1,-1)
Example 2
Identify functions
Tell whether each relation is a function. Explain.
a. Input
Output
b. Input
Output
2
1
3
6
1
0
4
2
1
2
2
0
2
3
Checkpoint Complete the following exercise.
1. Is the relation given by the ordered pairs (5, 2), (3, 1), (0, 0), (0, 2) and (0, 5) a function? Explain.
VERTICAL LINE TEST
A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.
Function
Not a function
Example3
Use the vertical line test
Is the relation represented by the graph a function? Explain.
a.
b.
GRAPHING EQUATIONS IN TWO VARIABLES
To graph an equation in two variables, follow these steps:
Step 1 Construct a table of values.
Step 2 Plot enough points from the table to recognize a pattern.
Step 3 Connect the points with a line or curve.
Example 4
Graph an equation in two variables
Graph the equation y = 2x  2.
Example 5
Graph the equation y  3
Example 6
Graph the equation x  2
Example 7
Classify and evaluate functions
Tell whether the function is linear. Then evaluate the function when x = 3.
a. f(x) = 6x + 10
b. g(x) = 2x2 + 4x 1
Checkpoint Complete the following exercises.
2. Use the vertical line test to tell whether the relation is a function.
3. Graph the equation y = 2x  3.
Tell whether the function is linear. Then evaluate the function when x = 1.
4. f(x) = 2x3 + 6  x
5. g(x) = 4x + 9
2.2 Find Slope and Rate of Change
Goal  Find slopes of lines and rates of change.
SLOPE OF A LINE
Words
The slope m of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run).
Algebra
y  y1

m 2
x2  x1
Example 1
Find slope
What is the slope of the line passing through the points (1, 3) and (6, 7)?
Let (x1, y1)  (1, 3) and (x2, y2)  (6, 7).
CLASSIFICATION OF LINES BY SLOPE
The slope of a line indicates whether the line rises from left to right, falls from left to right, is horizontal, or is
vertical.
Positive slope
Rises from left
to right
Negative slope
Falls from left
to right
Zero slope
Horizontal
Undefined slope
Vertical
Example 2
Classify lines using slope
Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical.
a.
(6, 2), b. (2, 1), (2, 2)
(1, 3)
Checkpoint Complete the following exercises.
1. Find the slope of the line passing through the points (4, 2) and (7, -9).
2. Without graphing tell whether the line through the points (3, 4) and (1, 4) rises, falls, is horizontal, or is
vertical.
SLOPES OF PARALLEL AND PERPENDICULAR LINES
Consider two different nonvertical lines l1 and l2 with slopes m1 and m2.
Parallel lines The lines are
parallel if and only if they have the
slope.
m1= m2
Perpendicular lines The lines are
perpendicular if and only if their
slopes are negative reciprocals
Example 3
Classify parallel and perpendicular lines
Tell whether the lines are parallel or perpendicular.
Line 1: through (3, 1) and (2, 5)
Line 2: through (3, 4) and (3, 1)
Checkpoint Tell whether the lines are parallel, perpendicular, or neither.
3. Line 1: through (1, 0) and (3, 4)
Line 2: through (24, 6) and (22, 5)
2.3 Graph Equations of Lines
Goal  Graph linear equations in slope-intercept or standard form.
PARENT FUNCTION FOR LINEAR FUNCTIONS
The parent function for the family of all linear functions is y = x. The graph of y = x is shown.
In general, a y-intercept of a graph is the y - coordinate of a point where the graph intersects the y-axis.
USING SLOPE-INTERCEPT FORM TO GRAPH AN EQUATION
Step 1 Write the equation in slope-intercept form by solving for y.
Step 2 Identify the y-intercept b and use it to plot the point (0, b) where the line crosses the y -axis.
Step 3 Identify the slope m and use it to plot a second point on the line.
Step 4 Draw a line through the two points.
Example 1
Graph an equation in slope-intercept form
3
Graph y =  x + 1.
2
USING STANDARD FORM TO GRAPH AN EQUATION
Step 1 Write the equation in standard form.
Step 2 Identify the x-intercept by letting y = 0 and solving for x. Use the x-intercept to plot the point where the
line crosses the x-axis.
Step 3 Identify the y-intercept by letting x = 0 and solving for y. Use the y-intercept to plot the point where the
line crosses the y -axis.
Step 4 Draw a line through the two points.
Example 2
Graph an equation in standard form
Graph 2x + 3y = 12.
Example 3
Graph horizontal and vertical lines
a. Graph y = 1
b. Graph x = 2.
Example 4
Graph the equation using any method
a. Graph 5 y  10 x  20
b. Graph 4 y  16  0
Checkpoint Graph the equation.
1. y = 2x + 2
4. 5x + 3y = 15
2. y =
4
x 4
3
5. y = 4
3. 4x + 2y = 8
6. 3x = 6
2.4 Write Equations of Lines
Goal  Write linear equations.
WRITING AN EQUATION OF A LINE
Use slope-intercept form: Given slope m and y-intercept b, use the equation y = mx + b.
Use point-slope form: Given slope m and a point (x1, y1), use the equation y – y1 = m(x- x1 ).
Use two points: Given points (x1, y1) and (x2, y2), first use the slope formula to find m. Then use the point-slope
form with either given point.
Write an equation given the slope and y-intercept
Example 1A
Example 1B
Write an equation of the line shown.
Write an equation of a line with m = 2/3 and b = 4
Example 2
Write an equation given the slope and a point
Write an equation of the line that passes through (2, 1) and has a slope of 2.
Example 3
Write equations of parallel or perpendicular lines
Write an equation of the line that passes through (1, 1) and is (a) parallel to, and (b) perpendicular to,
the line y = 2x + 3.
Write an equation given two points
Example 4A
Example 4B
Write an equation of the line
through (3, 1) and (2, 3).
Write an equation of the line
Example 5
Write an equation in standard form Ax +By = C of the line stat satisfies the given conditions. Use integers
for A, B, and C.
m = -3/2 and passes through (4, -7)
Checkpoint Write an equation of the line in y = mx + b form.
1.
3. Through (2, 3) and (a) parallel and
(b)
perpendicular to y = 4x  6
2. Through (1, 5) with a slope of 2
4. Through (6, 2) and (3, 2) leave answer in
Ax +By = C form
2.7 Use Absolute Value Functions and Transformations
Goal  Graph and write absolute value functions.
PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS
The parent function for the family of all absolute value functions is y = | x | . The graph of y = | x | is V-shaped
and is symmetric about the y-axis. So, for every point (x, y) on the graph, the point (x, y) is also on the graph.
The highest or lowest point on the graph of an absolute value function is called the vertex of an absolute value
graph. The vertex of the graph y  | x | is (0,0).
TRANSFORMATIONS OF GENERAL GRAPHS
For | a |  1, the graph is vertically stretched and y = a | x | is narrower than the graph of y = | x |.
For | a | < 1, the graph is vertically shrunk and y = a | x | is wider than the graph of
y  | x |.
Example 1
Graph functions of the form y = a | x |
Graph (a) y =
1
x and (b) y = 2 | x |. Compare each graph with the graph of y  | x |.
3
1
3
Checkpoint Graph the function. Compare the graph with y = | x |.
1. y = 3 | x |
Example 2
Graph a function of the form y  a | x  h |  k
Graph y = 3 | x  2 |  1. Compare the graph with the graph of y = | x |.
Example 3
Write an absolute value equation
Write an equation of the graph shown.
Checkpoint Complete the following exercises.
1
2. Graph the function y   | x  1 |  2. Compare the graph with the graph of
2
y = | x |.
3.
Write an equation of the graph shown.
2.8 Graph Linear Inequalities in Two Variables
Goal  Graph linear inequalities in two variables.
Example 1
Checking solutions of inequalities
Check whether the ordered pairs (a) (3, 2) and (b) (1, 4) are solutions of
4x + 2y > 6.
Checkpoint Check whether the ordered pair is a solution of 2x  y  8.
1. (6, 2)
GRAPHING A LINEAR INEQUALITY
To graph a linear inequality in two variables, follow these steps:
Step 1 Graph the boundary line for the inequality. Use a dashed line for < or > and a solid line for  or .
Step 2 Test a point not on the boundary line to determine whether it is a solution of the inequality. If it is a
solution shade the half-plane containing the point. If it is not a solution, shade the other half-plane.
Example 2
Graph a linear inequality with one variable
Graph y < 1 in a coordinate plane.
Example 3
Graph a linear inequality with two variables
Graph 3x  2y < 6 in a coordinate plane.
Example 4
Graph an absolute value inequality
Graph y > 3 x  1+ 2 in a coordinate plane.
Checkpoint Graph the inequality in a coordinate plane.
3.
x < 2
5. 9x + 3y > 9
4.
y  x + 2
6. y  2 |x + 2|  l