4.1 coordinates objective: to plot points and name points in the coordinate plane. a coordinate...
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4.1 Coordinates Objective: To plot points and name points in the coordinate plane.
A coordinate plane
An ordered pair
is formed by two real number lines that intersect at the origin. (x-axis and y-axis)
is a point in the coordinate plane represented by real numbers. The x-coordinate is the first number. The y-coordinate is the second number. Ex. (3,6)
(x, y) (right or left, up or down) How do you tell?
Coordinate plane (x, y)
x- axis
y-axis
Origin
(0,0)
Quadrant I
(+, +)
Quadrant II
(-, +)
Quadrant III
(-, -)
Quadrant IV
(+, -)
Plotting points
Then 4 up (positive) make a point
Plot these points:
1. (-2, -4)
2. (0, 3)
3. (-1,0)
4. (6,-2)
5. (-4, 5)
To plot a point: (3,4) Start at (0,0) Move 3 to the right (positive)
PracticeName the following points and give the quadrant or axis where they lie.
A:
B:
C:
D:
A
B
C
D
Plot the following points and label each!!
E (-3, 4) F(-5, 0) G(-3, -1) H(0, 0)
The Coordinate Plane
Steps to Make a Scatter Plot:1. Determine what will be x and y.
I. x – is in charge, it changes automaticallyII. y – depends on x, is not automatic
2. Determine units of each axis and label.I. Find range of variableII. Divide range by number of squaresIII. Always round up to “nice” unit
3. Plot points.
Make a Scatter PlotExampleThe age (in years) of seven used cars and the price (in thousands of dollars) paid for the cars are recorded in the table. Make a scatter plot and explain what it indicates.
0 1 2 3 4 5 6 78 9 10 11 12 13 14 150
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
age of carpr
ice
in $
1,00
0
Age 4 5 3 5 6 4 7
Price6.9
6.1
7.5
5.2
4.2
7.1
3.0
How much would a 2-year old car cost?
Make a Scatter PlotExampleThe amount (in millions of dollars) spent in the United States on snowmobiles is shown in the table. Make a scatter plot and explain what it indicates.
Year 90 91 92 93 94 95 96
Spent322
362
391
515
715
924
970
Reminders
•Math Lab Tomorrow - Stocks•4.1-4.3 Quiz on Monday, Oct. 7th
•Homework:• P. 206-207 #’s 10-26 EVEN, 35-37• EXTRA CREDIT – Halloween “Goblin’ Goblin” Grid
What is represented by this BrainBat?
TOOL O O O O LOOT
4.2 Graphing Linear Equations
Objective: To graph linear equations using a table of values.
The solution to a linear equation -- is an ordered pair (x, y).
There are many solutions to a linear equation and all of the solutions together form a straight line,
Note (1) All the Eqs. in Chap 4 refer 2 variable linear Eqs.
(2) The graph of each linear eq. is a LINE
Find out if the ordered pairs are solutions. HOW?
A) -5x – 8y = 15 (-3, 0) B) -2x – 9y = 7 (-1, -1)
Steps to graph a line
2. Plug in values for x then solve for y
3. Graph the ordered pairs
1. Pick three values for x
4. Connect these order pairs.
This should form a straight line!
y = -2x + 3
2. Solve for y, then evaluate y for all input x
x 2x + y = 3 y (x, y)
5
13
-1
1
0
Graph a line
Function form When an equation is solved for y =When an equation is solved for y =
What are the advantages of putting the equation into function form? Graph the given linear equations. A) 3x – y = 2
x y (x, y)
-1
1
0
Solve for y:
We select three x values, and evaluate the corresponding y values.
Graph the given linear equations.
B) 2x – 2y = 10 Solve for y,
We select three x values, and evaluate the corresponding y values. What are the “good” x values we should select? The x values should make “nice” y values. (or, no fraction values for y)
Make a table of values and graph the following line:
6x – 3y = 12 2y = 4x + 1
Special Linear EquationsSpecial Linear Equations
x = # x = #
y = # y = #
always a vertical linealways a vertical line
always a horizontal linealways a horizontal line
MEMORIZE THESE!!!! It’s easy!MEMORIZE THESE!!!! It’s easy!
Ex 4) y = -3 Ex 5) x = 4 Ex 7) y = 0Ex 6) x = -2
x = # , label the # on x-axis, then “cut” therex = # , label the # on x-axis, then “cut” there
y = # , label the # on y-axis, then “cut” therey = # , label the # on y-axis, then “cut” there
Make a table of values and graph the following line:
y = -3 x = 4
Summary1. A two variables linear equation represents a
line in x-y coordinate plan.
2. An ordered pair is a solution to a two variable linear equation, then the point represented by the ordered pair is on the line represented by the linear equation, and vice versa.
3. Remember the two types of special line by an easy way:
x = # no y cut x-axis at that # parallel to y-axis
y = # no x cut y-axis at that # parallel to x-axis
Summary
4. When graphing a linear equation, remember the 4 steps:
a) Pick a few x values
b) Solve for y, then evaluate y for all input x
c) Graph ordered pairs
d) Connect ordered pairs with a line
Lesson 4.2 DHQ
Decide whether the given ordered pair is a solution of 2x – 3y = 8.
a. (-2, -4) b. (7, -2)
Rewrite 4x – 2y = 18 in function form.
Reminders
•4.1-4.3 Quiz on Monday, Oct. 7th
•Homework:• P. 214-215 #’s 15-20, 30-32, 36-37, 60
What is represented by this BrainBat?
G G E G E G E G G
Where the line or curve crosses the x-axis. This should be written as the point (x, 0) . (WHY?)
4.3 Quick Graphs Using InterceptsObjective: To graph lines using x and y-intercepts.
What is an x-intercept?
What is a y-intercept? Where the line or curve crosses the y-axis. This should be written as the point (0, y) . (WHY?)
x-intercept? ( , )
y-intercept?( , )0 3
2 0
Note x or y intercept is a point!!!
6x + 3y = 12To find the y-intercept; let x = 0 and solve for y.
6( 0 ) +3y = 12
3y = 12
y = 4
The y intercept is (0, 4)
To find the x-intercept; let y = 0 and solve for x.
6x + 3( 0 ) = 12
6x = 12
x = 2
The x intercept is (2, 0)
Remember – TWO POINTS CAN MAKE A LINE!
Ex 1) Find the x and y-intercepts
1. 2x – y = 41. 2x – y = 4
You try these:
Calculate the x and y-intercept. Then graph each line.2. 3y – 2x = -62. 3y – 2x = -6
What happens with horizontal and vertical lines?
Find the x and y-intercepts (if possible). Graph each line.Ex 2) y = 4 Ex 3) x = -1
Variable x does not show up no x-intercept line is parallel to x-axis
Variable y does not show up no y-intercept line is parallel to y-axis
You can set x = 0 but end with “No solution” can not find x, or no x-intercept.
You can set y = 0 but end with “No solution” can not find y, or no y-intercept.
y-int. (0, 4) x-int. (-1, 0)
4. y = 2x + 44. y = 2x + 4
You try these:
Calculate the x and/or y-intercept. Then graph each line.5. y = -35. y = -3
6. Horizontal line 6. Horizontal line passing (-3, 4) and (4, 4). passing (-3, 4) and (4, 4).
Graph and Write the equation of the special line
7. Vertical line passing 7. Vertical line passing through (-2, 3)through (-2, 3)
Summary
1. x-intercept (y-intercept) is a point where the line or curve crosses the x(y)-axis.
2. To find x-intercept (y-intercept), just setting
y = 0 (x = 0) in an equation.
3. When graph a line, just find x and/or y-intercept and then connect two intercepts with a line.
Graphing with InterceptsWhat is an x-intercept?What is an x-intercept?
How do you find an x-intercept? How do you find an x-intercept? (Why does this work?)
How should you write the x-intercept?How should you write the x-intercept?
What is a y-intercept?What is a y-intercept?
How do you find a y-intercept? How do you find a y-intercept? (Why does this work?)
How should you write the y-intercept?How should you write the y-intercept?
Find the intercepts and graph.Find the intercepts and graph.3) 2x – 4y = –8 4) x – 4y = 2
So – If I gave you a quiz over this material, how would you do?
The point where line(curve) crosses the x-axis.
set y = 0 and then solve for x.
In an order pair (x, y).
The point where line(curve) crosses the y-axis.
set x = 0 and then solve for y.
In an order pair (x, y).
– 4y = –8y = 2
2x = –8
x = –4
– 4y = 2
y = – 1/2x = 2
Lesson 4.3 DHQ
1. Give the x- and y- intercepts of the graph of 2x – y = -4.
2. Graph 2x – 3y = 6 using x and y intercepts.
Reminders
•4.1-4.3 Quiz on Monday, Oct. 7th
•Homework:• P. 221-222 #’s 35-37, 44-49, 56-57
What is represented by this BrainBat?
C H I M A D E N A
runrisem)2
changehorizontal
changeverticalm)3
Objective: To calculate the slope of a line using 2 points, to read slope from a given line and to understand some applications of slope.
Slope is: 1) The measurement of the steepness and direction of a line (m)
1212xxyyxinchangeyinchangem)4
(ORDER MATTERS!)
To read the slope from a graph, choose 2 lattice points and write a ratio of the vertical to horizontal change. Explain.
Slope:
m =
m =
m =
4.4 The Slope of a Line
The slope formula – finding slopes from ordered pairs If you are given 2 points If you are given 2 points you have 2 x-values and you have 2 x-values and 2 y-values.2 y-values.
12
12xx
yym
1) (2, 4) and (3, -2) 2) (0, -5) and (1, 3)
3) (4, 5) and (4, -3) *** 4) (-2, -5) and (2, -5)
You need memorize it!!!
2 1
2 1
2 4
3 2
y ym
x x
66
1
5a) Sketch a positive slope.
b) Sketch a negative slope.
c) Sketch a 0 slope.
d) Sketch an undefined slope.
6) Describe how to move a slope of…….
a) -2 b)
c) d)
52
41
1
21
riserunm 2
5riserunm
14
riserunm
21, rise
runor m
14, rise
runor m
11
riserunm
25, rise
runor m
11, rise
runor m
Ex 5) The road rises 2 feet for every 50 yards. Find the slope of the road.
* REMEMBER TO CONVERT YDs to FT
Ex 4) The store plane descends 100 feet for every 2000 feet it travels.
Find the slope of decent. 100 1
2000 20
risem
run
2 1
50 25
risem
run
Summary
1. There are 3 different formulas to calculate the slope:
1)rise
mrun
2)vertical change
mhorizontal change
2 1
2 1
3)change in y y y
mchange in x x x
(order matters!!)
Lesson 4.4 DHQ1. Find the slope of each line. Use formula to find the slope.
a. (-2, 2), (0, 4)
b. (1, 1), (4, 2)
2. Find the slope of the line.
Reminders
• 4.4+4.6 Quiz on Wednesday, Oct. 16th
•Homework:• P. 230-231 #’s 23-28, 35-36 ***38
What is represented by this BrainBat?
sdraw
4.6 Graph Using Slope-Intercept Form
Objective: 1. To use slope intercept form to graph a line.
2. Tell the slope of two parallel lines.
Slope-Intercept Form is :
Memorize this form!!Memorize this form!!
1)1) The equation must be y = to use this short-cutThe equation must be y = to use this short-cut
2) m is always the coefficient of
3) y-intercept (b) is always
y = mx + b (m = slope & b = y-intercept.)y = mx + b (m = slope & b = y-intercept.)
xx
Constant – (no variable)Constant – (no variable)
1) 2x + y = -5
m= b =
2) 3y – 6 = 2x
m= b =
3) 2x – 5y + 10 = 0 4) 2x + 3y = -9
Solve the following equations for y = then pick out the m and b for each.
-2 -5
-2x -2xy = –2x - 5
Use the slope (m) and y-intercept (b) to graph each line.
2x + y = 5
m =
b =
x - 2y = 4
m =
b =
You need to graph
a) the y-intercept first and then
b) use the slope to get the second (third) point (move the slope)
c) connect the two (three) points by a line.
One more for you to try. 3x – 2y = –6
m =
b =
Can you solve for y? Can you locate the m and b?
Can you put this on the graph paper?
-3x -3x
3x – 2y = –6
Summary1. The slope-intercept form is the form in which y is
solved.
2. In the slope-intercept, the number in front of x is the slope m and the number right after x is the y-intercept.
3. When you graph a line of slope-intercept form, you have to graph
a) the y-intercept first and then
b) use the slope to get the second (third) point
c) connect the two (three) points by a line.
4. Parallel lines have equal slope.
Lesson 4.6 DHQ
1. Write x + y + 3 = 0 in slope-intercept form. Then graph the equation.
Reminders
• 4.4+4.6 Quiz on Wednesday, Oct. 16th
•Homework:• P. 244 #’s 13-18, 29-31
Where a line crosses the y-axis.
Organization Check:We can now graph an equation in three different ways:1.Using an x, y table2.Using x-intercepts, y-intercepts3.Slope-intercept form
Write the equation in slope-intercept form, then graph.
m =
b =
m =
b =
Write the equation in slope-intercept form, then
graph.
m =
b =
Write the equation in slope-intercept form, then
graph.
Write the equation in slope-intercept form, then graph.
m =
b =
Write the equation in slope-intercept form, then graph.
m =
b =
Write the equation in slope-intercept form, then graph.
m =
b =
m =
b =
Write the equation in slope-intercept form, then
graph.
Two different lines in the same plane are _______ if they do not intersect.
Lines are parallel if they have the same ______.
y = 2x + 3
y = 3 + 2x
y = -2x + 3
y = 3 + 2x
4x + 2y = 8
y = -2x + 2
Reminders
• 4.4+4.6 Quiz on Wednesday, Oct. 16th
•Homework:• P. 247 QUIZ 2 #’s 1-5, 13-14
If it takes six men one hour to dig six holes, how long does it take one man to dig half a hole?
Identify when a relation is a function.
4.8 Functions Objective: To evaluate functions at given values
Recall: Vocabulary
Relation:
Domain:
Range:
Function:____________________________________
Any set of points, equation or graph
The inputs ; the x-values from the points or graph
The outputs ; the y-values from the points or graph
A relation where all the inputs are different and any input has one and only one output
Ex. { (-3, 3), (1, 1), (3, 1), (4, -2) }
Domain:
Range:
Is this a function? Why?{-3, 1, 3, 4}
{3, 1, -2}Yes – all x-values are different
• For every input, there is exactly _____ output!one
Input 0 1 1 3
Output 3 1 3 2
Input 0 1 1 3
Output 3 1 1 2
Input 0 1 2 3
Output 3 4 5 6
•There are rules that associate more than one output for each input. But, we can’t call it a function anymore. We call it a…
•Any set of ordered pairs!•It doesn’t matter how many different outputs an input has!
Vertical Line Test – A relation is a function, if NO vertical line passes through two or more points!
FunctionRelation – Not a
function
Relation – Not a function
f(x) = 2x – 3 when x = -2, 0, and 3Steps:
1.Write the original function.
2.Substitute -2 for x.
3.Simplify.
g(x) = -5x + 3 when x = -3, 0, and 1
Graph f(x) = -2x + 3Steps for using slope-intercept form:
1.Rewrite the function as “y = …”
2.Find y-intercept and slope.
3.Graph and connect.
Graph f(x) = x + 4Steps for using intercepts:1.Find the x-intercept by setting y = 0.2.Find the y-intercept by setting x = 0.3.Plot the intercepts on the coordinate plane and graph the line.
Reminders
•Ch. 4 Test on Tuesday, Oct. 22nd
•Homework:• P. 259-260 #’s 16-17, 20-21, 37-40