Download Interactive Study Guide for Students: Trigonometric Functions

Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Section 1: Relations and Functions
Graph Relations
Examples
_______________: can be used to graph a relation of a series of two
numbers.
State the domain and range of
the relation shown in the graph.
Is the relation a function and
why?
Cartesian coordinate plane: composed of the x-axis and the y-axis
which meet at the origin (0,0) and divide the plane into four
____________.
1.
Relation: set of ordered pairs. The __________ is the set of all the
first coordinates (x-coordinates) from the ordered pairs, and the
______ is the set of all second coordinates (y-coor.)
__________: special type of relation where each element of the
domain is paired with exactly one element of the range. When the
relation has each element of the range being paired with exactly one
element of the range, it is called a one-to-one ____________
one-to-one
function(not one-to-one)
2.
not a function
The ____________ test can also be used to determine if it is a
function (f it intersects only one time, it is a function).
Graph the relation, find the
domain and range, and state if
it is a function and why?
3. y = 2x + 1
4. x = y2 -2
Equations of Functions and Relations
In a function, the ___________variable is the x or the domain. The
___________ variable depends of the domain.
______________ notation: Instead of y=2x+1, writef(x)=2x + 1
Interactive Study Guide for Students
5. Given f(x)=x2-2, Find f(3).
Chapter 2: Linear Relations and Functions
Section 2: Linear Equations
Identify Linear Equations and Functions
Examples
______equations: have no operations other than addition,
subtraction and multiplication of a variable by a constant.
Examples
Non-examples
5x – 3y = 7
7a + 4b2 = -8
y=½a
y=
x=5
State whether each function is
a linear function. Explain.
1. f(x) = 10 – 5x
2. g(x) = x4 -5
3. h(x,y) = 2xy + 1
x5
Write each equation in
Standard Form. Identify A, B,
and C.
2
x=
x
A linear __________ is a function that satisfies the linear equation, or
can be written f(x) = mx + b
4. y = -2x + 3
3
5
Standard Form
5. - x = 3y – 2
Ax + By = C where A  0, and A & B are not both zero
6. 3x – 6y -9 = 0
The y-coordinate of the point at which a graph crosses the y-axis is
called the y-________. Likewise, the x-coordinate of the point at
which it crosses the x-axis is the x-_________.
Find the x-intercept and the yintercept of the graph of 3x – 4y
+ 12 = 0.
Then graph the equation.
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Section 3: Slope
Slope
Examples
The _____________ of a line is the ratio of the change in ycoordinates to the corresponding change in x-coordinates. It
measures how steep a line is, and is often referred to as rate of
_____________. The slope m of the line passing through (x1, y1) and
(x2, y2) is given by:
1. Find the slope of the line
that passes through (-1, 4) and
(1, -2). Graph.
M = ___________ , where x1 ≠ x2
2. Graph the line passing
through (-4, -3) with a slope of
2
.
3
Parallel and Perpendicular Lines
A _______________of graphs is a group of graphs with one or more
similar characteristics. The _________graph is the simplest one.
3. Graph the line through (-1,
3) that is parallel to the line
with equation x + 4y = -4
__________ lines: non-vertical lines with the same slope.
______________ lines: two oblique lines with the product of their
slopes equal to -1. m1m2= -1 or m1 = -
1
m
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Section 4: Writing Linear Functions
Forms of Equations
Examples
_________-___________form: y = mx + b where m is the slope and
b is the y-intercept.
If you are given the coordinates of two points on a line, you can use
the ________-_________ form to find an equation of the line that
passes through them.
1. Write an equation in slopeintercept form for the line
that has a slope of -
3
and
2
passes through (-4, 1).
The point-slope form of the equation of a line is:
y – y1 = m(x – x1)
where (x1, y1)are the coordinates of a point on the line and m is the
slope of the line.
2. What is the equation of the
line through (-1, 4) and (-4,
5)?
3. As a salesperson, Eric is
paid a daily salary plus
commission. When his sales
are $1000, he makes $100.
When his sales are $1400, he
makes $120. Write a linear
equation to model this
situation.
4. Write and equation for the
line that passes through (-4, 3)
and is perpendicular to the
line whose equation is y=-4x-1
Parallel and Perpendicular Lines
The slope-intercept and point-intercept forms can be used to find
equations of lines that are parallel or perpendicular to given lines.
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Scatter Plots
Section 5: Using Scatter Plots
Examples
To model data with a function, it is helpful to
graph the data. A set of data graphed as
ordered pairs in a coordinate plane is called a
_________ _________.
1. The table shows the median selling price of new,
privately owned, one-family homes. Make a scatter plot
of the data
Yr
90
92
94
96
98
2000
$$ 122.9 121.5 130.0 140.0 152.5 169.0
Prediction Equations
When you find a line that closely
approximates a set of data, you are finding a
__________ of ____. An equation of such a
line is often called a ______________
____________ because it can be used to
predict one of the variables given the other
variable.
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Step Functions
2. The table show households that have a TV (in
millions) Draw a scatter plot, use two ordered pairs to
write a prediction equation, predict the missing value.
Yr
90
92
94
96
98
00
Houses 55
57
59
65
67
?
Section 6: Special Functions
Examples
A step function is not linear. It consists of line segments or rays. The
_________ ___________function, written f(x)= [[x]], is an example
of a step function. The symbol ________ means the greatest integer
less than or equal to x.
Constant Functions
In the function f(x)=b, when m=0, the value of the function is just
f(x)=b for every x value. So f(x)=b is called the ______________
function.
The Identity Function
Since the function f(x) = x does not change the input value, it is called
the _______________function.
The Absolute Value Function
A type of piecewise function is the Absolute Value Function. It is
called a ____________ function because it can be written using two
or more expressions
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Section 6: Special Functions
Step Functions
Examples
Parent Function: f(x) = [[x]]
1. f(x) = [[x + 3]]
x
y
D:
Domain:
R:
Range:
2. f(x) = 3[[x]]
Piecewise Functions
f(x) = x – 4 if x < 2
1
if x > 2
D:
x
y
R:
3. f(x) =
x+3 if x < -1
2x if x> -1
Domain:
D:
Range:
R:
The Absolute Value Function
Parent Function: f(x) = |x|
x
y
4. f(x) = |x-2| + 1
Domain:
D:
Range:
R:
Interactive Study Guide for Students
Chapter 2: Linear Relations and Functions
Section 7: Graphing Inequalities
Graph Linear Inequalities
The graph of a linear inequality resembles a linear equation, only it
acts as a boundary of the region.
Examples
1. 2x + 3y > 6
Step 1: Determine if line is solid or dashed.
Step 2: Check a point not on the line.
Step 3: If true; shade region of point.
If false: shade other side.
2. y < 2x - 1
Graph Absolute Value Inequalities
3. y < |-2x|
Graph similar to linear inequalities
Step 1: Determine if line is solid or dashed
Step 2: Check a point not on the line.
Step 3: If true; shade region of point.
If false: shade other side.
x
y
4. y > |x| - 2