2-1 Relations and Functions
Definitions
Relation – a set of ______.
e.g. {( 3, -5), (-2, 4), (0,7)}
Domain – the _____ values of a relation.
Range – the _____ values of a relation.
Function – a relation where each domain value is paired with exactly one
______value. For every x there is only one ____.
Does the relation make a function?
1. (2,-1),(3,3),(-4,1),(5,1),(1,1),(0,4)
2. (-1,-1),(3,5),(-4,1),(5,1),(3,1),(0,4)
Vertical Line Test
Use the Vertical Line Test to tell whether the graph of a relation is a function. If a vertical line passes
through at least ______points on the graph then the relation is ______a function (one element of
the domain is paired with more than one element of the range.)
Function Rules
2-2 Linear Equations
Graphing Linear Equations
Choose two values for x and find the ______values for y. Plot the points and draw the line.
example:
y = 3x – 4
If x = 0, y = -4 (0,-4)
If x = 1, y = -1 (1,-1)
Slope m =
Read the lines from left to right
Finding the Slope Given a Graph
- Find ______points on the line.
- Make a ______triangle with those two points.
- Put the ______on top, the ______on the
bottom.
- Check to see if the slope is ______,
if it is, add a ______sign.
- Reduce the fraction if you can.
Finding the Slope Given Two Points
Use the slope formula
If you are given ______points on the same line, you can use the slope formula to find the slope of the line going through them.
Find the slope of the line containing the points:
ex 2: (-2,1)(4,-3) ex 3: (-4,-3)(-1,5)
ex 4: (-2,-1)(-4,-1) ex 5: (2,-1)(2,-3)
Slope Intercept Form of the Equation of a Line
m is the ______, b is the ______
How to find the slope given a linear equation
Steps
1. Put the equation in ______form. (“y” by itself and positive)
2. The number in front of x, m, is the ______.
3. The other number (b) is the ______.
1) y = -4x + 8 2) -y = 3x – 7 3) 3x – 2y = 12
Finding the x-Intercept and the y-Intercept
x-intercept – where the line crosses the x-axis. At the x-intercept y = ___
y-intercept – where the line crosses the y-axis. At the y-intercept x = ___
1) y = -4x + 8 2) -y = 3x – 7 3) 3x – 2y = 12
Parallel and Perpendicular Slopes
Slopes of Parallel Lines
Two non-vertical lines are parallel if and only if their SLOPES ARE THE ______.
Slopes of Perpendicular Lines
Two non-vertical lines are perpendicular if and only if the product of their slopes equals -1. Their SLOPES
ARE ______RECIPROCALS.
Examples:
Slope of r = 3/5 Slope of a line perpendicular to r = -3/5
Slope of r = -3/2 Slope of a line perpendicular to r = 2/3
Slope of r = 5 Slope of a line perpendicular to r = -1/5
Fill in the rest of the table from the given information.
The Equation of a Line Standard Form and Point-Slope Form
Standard Form
A, B, C are real numbers, and A and B are not both zero.
Ax + By = C
example: 3x + 4y = 7; A = 3, B = 4, C = 7
Graph by finding the x- and y-intercepts
example: y – 6 = 3(x – 4); the point is (4,6), the slope is 3
Graph by using the point and the slope.
2-5 Absolute Value Functions and Equations
Definitions
Absolute Value Function Absolute Value Equation
f(x) = |mx + h| + k where m ≠ 0 y = |mx + h| + k where m ≠ 0
example: y = |3x + 2| - 5
Absolute Value Graph
To Graph an Absolute Value
start by finding the Vertex:
example: y = |2x - 4| + 3,
vertex =
then chose two values for x, one less than and
one greater than the vertex’s x-value.
Put them in a table, plug them in, and find the y-value.
2-6 Translating Absolute Value Functions
Absolute Value Translations
Vertical Translation
Parent Function: y = | x |
Up k units, k > 0: y = | x | + k
Down k units, k > 0: y = | x | - k
a) Graph the parent function.
b) Graph y = | x | - 4
c) Graph y = | x | + 6
Horizontal Translation
Parent Function: y = | x |
Right h units, h > 0: y = | x - h |
Left h units, h > 0: y = | x + h |
a) Graph the parent function.
b) Graph y = | x - 4 |
c) Graph y = | x + 6 |
Combined Translation
Right h units, up k units:
y = | x - h | + k
a) Graph y = | x - 4 | + 2
b) Graph y = | x + 6 | - 3
Stretch and Shrink
Stretch – a vertical stretch multiplies all
____-values by the same factor greater than _____.
example: Graph the parent function y = | x |,
then stretch it by graphing y = 2| x |.
Shrink – a vertical shrink reduces ____-values
by a factor between ______.
example: Graph the parent function
y = | x |, then shrink it by graphing
y = ½ | x |.
2-7 Two-Variable Inequalities
Linear Inequality
y > mx + b
Graphing Hints:
dotted line, shade above the line
≥ solid line, shade above the line
dotted line, shade below the line
≤ solid line, shade below the line
Examples
Graph
1. 3x – 2y ≤ -4
2. x < 2
3. y > -2
4. x ≥ y