Download Linear Functions

Linear Functions
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Graph is a straight line
General form: y = mx + b or, equivalently, f(x) = mx + b, where m, b are constants
Alternate form is Ax + By = C, where A, B, and C are constants
Linear
m
b
y=3 x – 7
2
f ( x) = − x
3
3 x − 5 y = 10
Non-linear
Reason
y=x 2−5
f ( x)=
5
x−3
g(t)=√ t
Graphing Linear Equations using a Table of Values
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Ex1:
You need a minimum of _____ points to graph a linear function, but ______ points is better.
Choose values for x that will make your life easy! Always choose x = 0 !
2
Graph f (x) = − x
3
Ex2: Graph 2x - 5y = 10
a. Solve for y in terms of x
b. Table of Values:
Intercepts
An x-intercept of a graph is a point at which the graph intersects the __________________.
A y-intercept of a graph is a point at which the graph intersects the __________________.
For the graph to the left, the
x-intercepts are: ______________________________
and the y-intercept is:
_________________________
Note: The zeros of this graph are _____ and _____ , the same as
the x-intercepts.
Computing Intercepts
To find x-intercept(s), set ________________ and solve for x. This also gives you the zeros of f(x).
To find the y-intercept, set ________________ and solve for y.
Example: Compute the x and y intercepts of f(x) = -2x + 7
This function has one zero, and it is at ______________.
Graphing Linear Functions using the Intercept Method
Since you only need two points to construct the graph of a line, you could just use the intercepts.
Example: Graph 2x – 5y = 10 using the intercept method:
You try it: Graph 3x + 4y = 8 using the intercept method:
Horizontal and Vertical Lines
Horizontal Line – general form is y = b
Vertical Line – general form is x = a
On a horizontal line, all of the
On a vertical line, all of the
_____ coordinates are the same.
____ coordinates are the same.
Application Problem – Depreciation
An company purchases a piece of equipment for $5,000 and expects it to lose approximately $800 in
value per year. The formula for the predicted value of this piece of equipment is:
V(t) = -800 t + 5000, where V is the value (in dollars) after t years.
1. Compute V(0) and V(5) and graph
2. Explain the meaning of V(5) in words.
3. Compute the zero of V(t). What does this tell you about this piece of equipment?
Slope of a Line
The slope of a line (m) is a mathematical way to describe the pitch (or steepness) of a line.
Positive slope
uphill
m≈½
Positive Slope
(larger value)
m≈2
Negative Slope
downhill
m ≈ -1.5
Zero slope
flat
m=0
There are many ways to remember slope:
m=
rise
run
or
m=
change in y
change in x
or
m=
Δy
Δx
The formula for the slope of a line containing points P1 (x 1, y 1 ) and P2 (x 2, y 2) is:
m=
y2 − y1
x2 − x1
Problem 1: Find the slope of the line containing (2, -3) and (-1, 1).
The rise is ____________ and the run is __________ to the right.
Problem 2: Find the slope of the line containing (3, 2) and (1, 7).
The rise is ____________ and the run is __________ to the right.
Undefined
slope
Problem 3 – Find the slope of the line containing (3,5) and (3, 10)
The slope of this line is ______________________ because the line is ___________________.
BTW - the equation of this line is ______________________________.
Graphing a Line using Slope and Intercept (y = mx + b)
An equation of the form y = mx + b is in slope-intercept form, because m = slope and b = y-intercept.
We can use this information to quickly and easily graph an equation in this form:
Ex1: Graph y = 3x – 4
1. Start at the y-intercept, which is __________
2. From this point, move up _____ and over ______,
because the slope is _______. Put another point here.
3. Repeat (optional, but recommended). Draw the line.
Ex2: Graph
f (x) = − 32 x +1
1. Start at the y-intercept, which is __________
2. From this point, move down _____ and over ______,
because the slope is _______. Put another point here.
3. Repeat (optional, but recommended). Draw the line.
Ex3 – You try it: Graph 4x – 3y = 9 using the slope/intercept method:
Finding Equations of Lines
Objective: Find the equation given a point and the slope:
Ex1: Use y = mx + b to find the equation of a line given m = 2 and a point (3, -4).
The goal is to figure out both m and b. Final equation is:
Alternate Approach – using the point slope form of a line
y = mx + b is called the ___________________________________ form of a line. An alternate form,
useful when creating equations from data, is the point-slope formula:
y− y 1=m(x −x1 )
where (x 1, y 1)
is any point on the line.
Let's try the same problem as before using this formula:
Ex1: Given m = 2 and the point (3, -4), find the equation of the line.
Ex2: Given m = -3/2 and the point (-4, 6), find the equation of the line.
Note: you can use either point-slope, slope-intercept, or just
common sense to find equations of lines:
Ex3: Find the equation of the line passing through (3, 4)
where the slope is undefined.
Ex4: Find the equation of the line passing through (-1, -4) and (2, -8).
Strategy, compute the _____________ first, then use _________________________ as (x 1, y1 )
Ex5: Find the equation of the line containing the points (5, 2) and (5, 4).
Application Problems - Try to determine at least two pieces of information (either one point and the
slope, or two points) to construct the equation. Once you are done, answer any questions.
c. What is the base salary?
Independent Variable (x) is _______________________________
Dependent Variable (y) is ________________________________
( x 1, y1 ) equals:
(x 2, y 2) equals:
Independent Variable (x) is _______________________________
Dependent Variable (y) is ________________________________
Parallel and Perpendicular Lines
If you look at the parallel lines to the right, you can see that
the slopes of the lines are _______________ to each other.
Similarly, it can be shown that perpendicular lines
have slopes that are ______________________ ________________________ of each other.
Fill in the chart:
Equation of a line
----------------------
slope of this
line (m)
----------------------
slope of any
parallel line - //
---------------------
slope of any
perpendicular line - ⊥
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y = 4 x +7
3
y= − x+1
2
2 x – 5 y=10
y=3
Finding equations
Ex1: Find the equation of the line parallel to y = 5x + 2 passing through (3, -1).
Ex2: Find the equation of the line perpendicular to 2x + 3y = 1 with an x-intercept of (12, 0).
Graphs of Linear Inequalities in Two Variables
Consider this inequality in two variables: y > 2 x−3 . We know how to
graph the related equation in two variables y = 2 x−3 as follows:
2
, b = -3 (y-intercept)
1
•
m=2=
•
start at (0, -3), move ____________ and _________________ to
find the next point on the line.
Now, back to y > 2 x−3 . Does the point (0, -3), which was part of the
solution set above, work in this inequality?
We represent this fact by graphing with a ________________ line.
Finally, it can be shown that the solution to an inequality in two-variables is
a half-plane, which is one side of the line or the other.
Pick a point on either side of the line, use (0, 0) when you can. Test it in
y > 2 x−3 ... 0 > 2(0) - 3 ... 0 > -3 is this TRUE or FALSE.
If TRUE, shade the side containing (0,0).
If FALSE shade the other side.
Notes on linear inequalities in two-variables
1. Given an inequality, graph the related equation.
2. Use a _________________ line for
<
or
>, and a ________________ line for ≤
3. Pick a test point, (0, 0) if available, and plug into the inequality.
◦ if the result is a TRUE statement, shade towards the test point.
◦ if the result is a FALSE statement, shade the other way.
Ex1: Graph
y ≤ − 12 x − 1
Graph __________________ using a ____________ line
because of _____.
Then test the point _________ in
y ≤ − 12 x − 1
Shade ____________________ the point ________ .
Ex2: Graph 3 x - 2 y > 6.
or
≥