3 - N1 COORDINATE PLANE
Today, you will be able to:


When graphed, a LINEAR EQUATION will form a _________________.
How can you tell if an equation is a linear equation?
1.
2.
Which of the following is a linear equation?
(1)
y = x 2 + 3x - 5
(3)
y < 2x +8
(2)
y = x3 + 8
(4)
y = 2x +8
Which graph represents a linear equation?
1)
3)
2)
4)
VERIFYING A POINT AS A SOLUTION:
3.
Given the following linear graph:
Is the point ( 0, 0) a solution?
Is the point ( 4, 2) a solution?
Is the point (-5, -1) a solution?
Explain your answer based on the graph.
SLOPE-INTERCEPT FORM OF A LINEAR EQUATION:
where m represents the ____________________________
and b represents the ____________________________.
However, instead of writing y = mx + b , we can use function notation instead!
FUNCTION NOTATION:
4.
The graph of y = f ( x ) is shown below.
Which point could be used to find f ( 2) ?
Which point could be used to find f ( -2 ) ?
Evaluate f ( 0) .
Evaluate f ( -1) .
5.
The graph below represents a jogger’s speed during her 20-minute jog
around her neighborhood.
Evaluate f (8) .
Evaluate f (15) .
If f ( x ) = 8 , then what could x be?
If f ( x ) = 4 , then what could x be?
3 – N2 GRAPHING LINES
Today, you will be able to:

GRAPHING HORIZONTAL AND VERTICAL LINES:
To graph x = #:
To graph y = #:
1.
Graph x = 4.
2.
Graph y = -2.
3.
Graph x = -5.
4.
Graph y =1.
GRAPHING USING SLOPE-INTERCEPT FORM:
Use the following trick:
When graphing:
Rise
Run
Graph the following lines:
5.
y=
1
x +2
3
6.
1
f ( x) = - x - 3
2
9.
y=
8.
f ( x) = x
10.
f ( x) =
2
x +2
3
3
x
2
11.
f ( x ) = 2x + 3
12.
f ( x) = x - 2
13. y = -3x
7. y = -2x +1
14.
f ( x ) = -x + 4
15.
y=
5
x-4
2
16.
3
f ( x) = x
5
3 – N3 RATE OF CHANGE
Today, you will be able to:


The steepness of a line is called its ___________________.
We can also call this the _______________________________.
The formula for RATE OF CHANGE is:
1.
Graph y = - 1 x - 4 .
2
What is the average rate of change over -4 £ x £ -2 ?
What is the average rate of change over 0 £ x £ 2 ?
What is the average rate of change over 2 £ x £ 4 ?
In a linear equation, the average rate of change is _________________.
2.
Find the average rate of change.
3.
Find the average rate of change.
Now, what is the equation of the line?
When given a graph, you will need to find the rate of change and the
y-intercept!
4.
______________ 5.
_______________ 6.
_______________
7.
______________ 8.
_______________ 9.
_______________
10.
______________ 11.
_______________ 12. _______________
3 – N4 FUNCTIONS: Domain and Range
Today, you will be able to:


When you are given a graph, a FUNCTION passes the
__________________________________________________.
Any vertical line you draw can intersect the graph at only ______________.
1.
Which graph does not represent a function?
2.
Which graph represents a function?
DOMAIN:
RANGE:
3.
The accompanying graph shows the elevation of a certain region in
New York State as a hiker travels along a trail.
What is the domain of this function?
What is the range of this function?
4.
The effect of pH on the action of a certain enzyme is shown in the
accompanying graph.
What is the domain of this function?
5.
The accompanying graph shows the heart rate, in beats per minute, of
a jogger during a 4-minute interval.
What is the range of the jogger’s heart rate during this interval?
6.
(1)
0<x<4
(3)
60 < y <110
(2)
0£x£4
(4)
60 £ y £110
Graph y = x - 4 .
What is the domain of this function?
What is the range of this function?
3 – N5 FUNCTIONS: One-to-One
Today, you will be able to:



ONE-TO-ONE FUNCTION:
1.
2.
Function?
Domain:
One-to-One?
Range:
Function?
Domain:
One-to-One?
Range:
Function?
Domain:
One-to-One?
Range:
3.
4.
Function?
Domain:
One-to-One?
Range:
NATURE OF THE GRAPH:
Increasing:
Decreasing:
INTERCEPTS:
x-intercept:
y-intercept:
5.
Graph the line that contains the following points:
x
y
-6
5
-4
4
-2
3
0
2
Domain:
Range:
Function?
One-to-One?
Nature of the Graph:
x-intercept:
y-intercept:
6.
Graph the following relation:
y = x+5
Domain:
Range:
Function?
One-to-One?
Nature of the Graph:
x-intercept:
y-intercept:
3 – N6 Word Problem Equations
Today, you will be able to:

1.
Super Painters charges $1.00 per square foot plus an additional fee of
$25.00 to paint a living room. If x represents the area of the walls of
Francesca’s living room, in square feet, and y represents the cost, in
dollars, which graph best represents the cost of painting her living
room?
2.
Madison wants to take a spinning class. With her student discount, she
will be charged a $6 sign-up fee, plus an additional charge of $2 per class.
Model this situation by using a linear function. Use c(x) to represent
the total cost, and x to represent the number of classes.
Graph the function.
Using your graph, determine Madison’s total cost if she decides to
attend five classes.
3.
A piano is being lowered from an apartment that is 18 feet above the
sidewalk. The piano descends at a constant rate of 3 feet per minute.
Model this situation by using a linear function. Use h(x) to represent
the height of the piano in feet, and x to represent the time in minutes.
Graph the piano’s elevation over the domain 0 £ x £ 6 .
Using your graph, determine the amount of time it takes for the piano
to reach the sidewalk.
4.
Sean is at his grandmother’s house, which is 24 miles from his home.
He starts riding home at time t = 0. His distance from home, d, after t
hours can be modeled by the function d(t) = 24 - 6t . Graph the
function for the domain 0 £ t £ 4 .
Explain why the domain must be restricted in this way.
What does the y-intercept represent in this situation?
What does the x-intercept represent in this situation?
3 – N7 SYSTEMS OF LINEAR EQUATIONS
Today, you will be able to:

SYSTEMS OF EQUATIONS:
SOLUTION TO SYSTEMS OF EQUATIONS:
TO SOLVE A SYSTEM OF LINEAR EQUATIONS GRAPHICALLY:
1.
y = x +1
y = -x + 5
3.
x=4
y = -2
4.
y = 2x + 5
y=x+4
y = -2x - 3
2.
y=
1
x +2
2
5.
6.
y=7
y = x+4
1
y=- x
2
1
y =- x+4
2
7.
y = 3x +1
y = -5x + 9
8.
2
y = - x -2
3
7
y =- x+3
3
3 – N8
TRANSLATE SYSTEMS FROM WORD PROBLEMS
Today, you will be able to:

1.
A baker rents a space in a commercial kitchen for $24. For each pie
he bakes, he spends $2 on materials. Model the baker’s costs by
using a linear function. Use c(x) to represent the cost, in dollars, and x
to represent the number of pies.
The baker charges $10 per pie. Model the baker’s revenues by using a
linear function, using r(x) to represent the revenue, in dollars, and x to
represent the number of pies.
Using an appropriate scale, graph the baker’s costs and revenues.
How many pies must he sell in order to break even?
2.
Sonya and Brad started saving their allowances. Sonya began with
$26 and contributes $2 each week. Brad began with $8 and
contributes $5 each week. Write a linear equation to represent
Sonya’s savings, s(w), and Brad’s savings, b(w), after w weeks.
Sonya:
Brad:
Using an appropriate scale, graph Sonya and Brad’s functions.
After how many weeks will Brad’s savings be the same as Sonya’s?
What will the total savings be then?
3.
Rachel goes trick-or-treating for Halloween. She collects 40 pieces of
candy and she decides to eat 2 pieces of candy per day. Write an
equation to model Rachel’s candy remaining using r(d) to represent
Rachel’s remaining pieces of candy, and d to represent each day.
Madison also goes trick-or-treating and she collects 49 pieces of
candy. She decides to eat 5 pieces of candy per day. Write an
equation to model Madison’s candy using c(d) to represent Madison’s
candy, and d to represent each day.
Using an appropriate scale, graph Rachel’s candy and Madison’s candy.
One what day will both girls have the same amount of candy
remaining?
3 – N9 TRANSFORMATIONS
Today, you will be able to:

1.
Solve the equation for x:
4x -10 = x -1
We can also solve equations graphically!
2.
Solve the equation for x:
Now solve graphically:
5
x -1 = -6
2
The graphs of all functions in the FAMILY OF LINEAR FUNCTIONS
are __________________________.
Each family of functions has a __________________________________,
the most basic function in the family.
The family of linear functions has the parent function _________________.
A change to the equation of a function causes a change to the graph.
This change to the graph is called a _______________________________.
VERTICAL TRANSLATION:
If k > 0, shift the parent function __________________.
If k < 0, shift the parent function ___________________.
3.
Graph the parent function of the family of linear functions, y = x.
Now, on that same set of axes, graph the function y = x + 3.
Now try y = x - 3.
These new graphs are called vertical translations!
4.
1
2
Graph the function y = - x +1.
Graph the function created by a vertical translation 5 units down.
What is the equation of this new function?
5.
3
2
If you were to graph the function y = - x -1, and then you performed
a vertical translation 7 units up, what would be the equation of this
new function?
6.
If you were to graph the function y = -x + 4, and then you performed a
vertical translation 10 units down, what would be the equation of this
new function?