UNIT 3:
LINEAR FUNCTIONS
What is a linear function?
It
is a graph of a LINE, with 1 dependent variable
or output or y and 1 independent variable or
input or x.
The
rate of change or slope is the SAME for ANY 2
points on the line.
There
are infinitely many points on a line!!
Rate of Change and Slope
Rate
of Change = change in dependent variable
change in independent variable
Slope = vertical change = run
horizontal change
rise
Slope =
y 2 - y1
x 2 - x1
y1 - y 2
= x1 - x 2
or x 1 - x 2 CANNOT be 0 or slope is undefined
(vertical line)
x 2 - x1
What is a linear function?
4
types of line, but ONLY 3 qualify as a function
(1 unique output for 1 input)
Positive Slope
Negative Slope
Zero Slope
Horizontal
Line
Undefined Slope
NOT
a function
Does not pass vertical line test!!
X=3 is a linear equation NOT a linear function
Problems/Need to Know
Identifying
positive, negative, zero, and
undefined slope graphs.
Find
slope given 2 points.
Find
x or y coordinate given slope and points:
Slope=3, points (1,2) and (2,y) lie on the line, find y
Slope=2, points (3,2) and (x, 10) lie on the line, find x
Find
the slope or rate of change given a
graph/plot
x and y intercepts
x intercept is the POINT where the graph crosses the xaxis or when the value of y is 0
y intercept is the POINT where the graph crosses the yaxis or when the value of x is 0.
Examples (0,0), (3, 0), (-6,0), (-4,0), (12,0)
Examples (0,0), (0, 2), (0,10), (0,-3), (0,-10)
Is there an x and y intercept for:
Positive slope line?
Negative slope line?
Yes, will have both x and y intercepts
Yes, will have both x and y intercepts
Zero slope line?
Will have ONLY y-intercept. Remember 0 slope = horizontal line
Direct Variation
Graph
of a line with a y-intercept of ZERO, meaning
the graph passes through the ORIGIN.
Direct
variation is defined by an equation of the
form y=kx, where k is the constant of variation or
the slope of the line.
Joint Variation
Joint
y = kxz
variation: y = kxz
Ex.
y varies jointly with x and z
1
A
=
bh . The area A varies jointly
Ex. Area of Triangle,
2
with length of the base b and height h where the
constant of variation k is equal to ½.
The
graph of a joint variation also has a y-intercept
of ZERO, meaning the graph passes through the
ORIGIN.
k
is the again the constant of variation or the slope
of the line.
Joint Variation
y = kxz
Problems/Need to Know
Identifying the x and y intercepts on a graph/plot
Find the x and y intercept given a linear equation
Identifying whether a linear equation is a direct
variation, if it is, what is the constant of variation k
Ex. Find the x and y intercept for y=2x-10
Ex. Find the x and y intercept for 3x-4y=-20
Ex. Is this a direct variation y=3x, yes, k=3
Ex. Is this a direct variation 2x-3y=10, NO
Ex. Is this a direct variation -3x+2y=0, yes, k=3/2
Find the value of k (constant of variation) for both
direct and jointly variation problems and use k to find
missing value
Slope Intercept Form
y = mx + b
m is your slope
b is your y-intercept or the y-coordinate of your y-intercept
To write an equation in slope intercept form, you need the
slope m and b, your y-intercept.
Ex. Given a line with a slope of 2 passes through the point
(0,5). The equation for the line in slope intercept form is
y = 2x + 5
There is ONLY 1 slope intercept form for a linear function.
WHY??
Any point (x, y) on the line MUST satisfy the equality with slope
and y-intercept.
Point Slope Form
y - y 1 = m (x - x 1 )
To
write a linear equation in Point Slope Form, you
need
Slope m
Any point on the line, (x 1, y 1 )
Ex.
Given a line with a slope of -3 passes through
the point (2,-3), the equation in point slope form is
y + 3 = - 3(x - 2)
Infinite
number of point slope form equations for a
line: WHY??
Standard Form
A,
A
A x + By = C
B, and C are real numbers.
Real number is any number on a number line; pretty
much any number you can think of.
and B CANNOT both be zero. Why??
Standard
form allows finding the x and y intercepts
of a linear function easier.
Ex.
2x+3y=6
Problems/Need to Know
Given
two points on a line: write the equation in
slope intercept form, point slope form, standard
form, and graph.
Ex. Given (2,5) and (-2,-7)
Given
slope and 1 point on a line: write the
equation in slope intercept form, point slope form,
standard form, and graph.
EX. Given m=-2 and a point (1,7)
Given
stand form for a linear function, solve for y
and write it in slope intercept form.
Ex. Given 3x-5y=10, write it in slope intercept form
Problems/Need to Know
Given
a table of x and y values: determine if it is a
linear function and if yes, write the equation in
slope intercept form, point slope form, standard
form, and graph.
Determine
equation.
whether a point is a solution to a linear
Ex. Is (3, 5) a solution to the linear equation 2x-3y=-9?
Rewrite
a slope intercept form equation with
fractions in standard form using integers
Ex. Rewrite linear equation
form using integers
y=
2
x- 4
3
in standard
Interpreting Real Life Examples/Situations
of Linear Relationship
Ex.
I already made 5 baskets to sell at the fair.
After working for another 3 hours, I made 6 more
baskets for a total of 11 to sell at the fair. Write a
linear equation relating the number of baskets
made (y) to the amount of time spent (x) making
the baskets. How many baskets total will I have for
the fair if I spend another 7 hours on making
baskets?
Ex. A hot air balloon descends at a linear rate of 2
meters per second from a height of 600 meters
above ground. Write the linear equation for the
descend? What does the x and y intercept
mean?
Parallel and Perpendicular Lines
Parallel Lines: lines that never intersect.
Perpendicular Lines: lines that intersect to form a 90
degrees or right angle.
Slope: nonvertical parallel lines have the same slope but
different y-intercept. Vertical parallel lines are parallel if
they have different x-intercept.
Slope: Two nonvertical lines are perpendicular if the
product of their slopes is -1. A vertical line and a horizontal
line are also perpendicular.
Opposite reciprocals: two numbers whose product is -1.
Ex. ½ and -2, -3/4 and 4/3, 5 and -1/5
Parallel and Perpendicular Lines
Problems
Write
an equation of a line passing through point
(12, 5) and is parallel to the graph of y=2/3x-1.
Know: slope
Need: the y-intercept for y=mx+b
Classifying
Lines: are the graphs of 4y=-5x+12 and
y=4/5x-8, parallel, perpendicular, or neither
Slope of second line is 4/5
Slope of first line can be found by solving the
equation for y which comes out to be -5/4
They are opposite reciprocals: so the lines are
perpendicular.
Parent Function: See Notes/Handout
We will learn the following 3 families of functions:
Parent Linear Function:
y= x
Equation of lines
2
Example of linear functions: y = 2x - 1, y = - 3x + 10, y = x + 3
3
Parent Quadratic Function:
y = x2
Equation with a variable x raised to the 2nd power
Examples of quadratic function:
y = 3x 2 , y = - 2x 2 - 1, y = 4x 2 + x - 3
Parent Absolute Value Function:
y = x
Equations with an absolute value symbol around x
Examples of absolute value functions:
y = 2 x - 1, y = x - 3 , y = 3 x + 4
Absolute Value: l l
Distance from 0.
NO NEGATIVES
Absolute value of ANYTHING is POSITIVE!
Simplify | 2 + 3(–4) |.
| 2 + 3(–4) | = | 2 – 12 | = | –10 | = 10
Simplify –| –4 |.
–| –4| = –(4) = –4
Simplify –| (–2)2 |.
–| (–2)2 | = –| 4 | = –4
Simplify –| –2 |2
–| –2 |2 = –(2)2 = –(4) = –4
Simplify (–| –2 |)2.
(–| –2 |)2 = (–(2))2 = (–2)2 = 4
3-7 Transformation of Linear Functions
Transformation
of a function is a change to the
equation of the function that results in a change in
the graph of the function.
**Graph
of ANY linear function is a transformation
of the parent function y(x)=x.
Translation:
a vertical or horizontal shift in a graph.
Reflection:
flip the graph of a function across the x
or y axis
3-7 Transformation of Linear Functions
3-7 Transformation: Translation
Slope
DOES NOT
change.
Adding
a value d to the
outputs shifts the graph
up. y= f(x) + d
a positive
value of c from the
inputs shifts the graph to
the right. y= f(x-c)
BLUE : f (x ) = x
R ED : y = f (x ) + 5 = x + 5
Subtracting
BLUE : f (x ) = x
R ED : y = f (x - 3) = x - 3
3-7:Translation
More
Examples:
Adding
a value d to the
outputs shifts the graph
up. y= f(x) + d
BLUE : f (x ) = 2x - 3
R ED : y = f (x ) + 5 = (2x - 3) + 5 = 2x + 2
Subtracting
a positive
value of c from the
inputs shifts the graph to
the right. y= f(x-c)
BLUE : f (x ) = 2x - 3
R ED : y = f (x - 3) = 2(x - 3) - 5 = 2x - 9
3-7 Transformation of Linear Functions
3-7 Transformation of Linear Functions
3-7 Linear Transformation Vocabulary
X-intercept
increases or graph shifts to the right
X-intercept decreases or graph shifts to the left
Y-intercept increases or graph shifts up
y-intercept decreases or graph shifts down
Reflected
over the x-axis
Reflected over the y-axis
Steeper:
slope increases
Flatter: slope decreases
Parallel: slope stays the same
Scatter Plots and Trend Lines
Scatter
Plot: a graph the relates to different sets of
data by displaying them as order pairs.
Trend
line: a line on a scatter plot, drawn near the
points, that show a correlation.
Correlation:
variables
statistical relationship between 2
Positive correlation: y tends to increase as x increases
Negative correction: y tends to decrease as x
decreases
Zero correlation: two sets of data are not related
Scatter Plots and Trend Lines
Scatter Plots and Trend Lines
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Problems/Need to Know
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