Transcript Expressions, Rules, and Graphing Linear

Expressions, Rules, and
Graphing Linear
Equations
by Lauren McCluskey
• This power point could not have been made
without Monica Yuskaitis, whose power point,
“Algebra I” formed much of the introduction.
Variable:
• Variable – A variable is a letter or
symbol that represents a number
(unknown quantity or quantities).
• A variable may be any letter in the
alphabet.
• 8 + n = 12
“Algebra I” by M. Yuskaitis
Algebraic Expression:
• Algebraic expression – a group of
numbers, symbols, and variables that
express an operation or a series of
operations with no equal or inequality
sign.
• There is no way to know what quantity or
quantities these variables represent.
•m+8
•r–3
“Algebra I” by M. Yuskaitis
Simplify
• Simplify – Combine like terms and
complete all operations
m=2
•m+8+m
• 3x + (-15) -2x + 5
2m+8
x -10
“Algebra I” by M. Yuskaitis
Evaluate
• Evaluate an algebraic expression – To find
the value of an algebraic expression by
substituting numbers for variables.
•m+8
•r–3
m = 2 2 + 8 = 10
r=5 5–3=2
“Algebra I” by M. Yuskaitis
Translating Words to
Algebraic Expressions
•
•
•
•
•
Sum
More than
Plus
Increased
Altogether
Difference
Less than
Minus
Decreased
“Algebra I” by M. Yuskaitis
Translate these Phrases to
Algebraic Expressions
• Ten more than a number
• A number decrease by 4
• 6 less than a number
• A number increased by 8
• The sum of a number & 9
• 4 more than a number
n + 10
n-4
x-6
n+8
n+9
y+4
“Algebra I” by M. Yuskaitis
Each of these Algebraic
Expressions might represent
Patterns:
• For example: n + 10
(x)
(y)
1
11
2
12
3
13
Or it might be Geometric (n-4):
n=1
13 seats
n=2
8 seats
Patterns
Patterns are predictable.
Patterns may be seen in:
• Geometric Figures
• Numbers in Tables
• Numbers in Real-life Situations
• Sequences of Numbers
• Linear Graphs
Patterns with Geometric
Figures (Triangles)
• Jian made some designs using
equilateral triangles. He noticed that as
he added new triangles, there was a
relationship between n, the number of
triangles, and p, the outer perimeter of
the design. Write a rule for this pattern.
P= 4
P=3
P=6
P=5
from the MCAS
How to Write a Rule:
1) Make a table.
2) Find the constant difference.
3) Multiply the constant difference by
the term number (x).
4) Add or subtract some number in
order to get y.
P=6
P=4
P=3
P=5
1 ) Make a Table:
Let x be the position in the pattern while y
is the total perimeter.
# of Triangles Rule:
Perimeter
(x)
(y)
1
2
3
...
x
?
3
4
5
…
y
from the MCAS
P=4
P=6
2)Find
the Constant Difference:
How did the
P=3
P=5
output change?
Perimeter
(y)
3
4
5
6
…
p
+1
+1
+1
from the MCAS
P=4
3) Multiply by the Input # (x).
P=3
P=5
4) Then Add or Subtract some # to get
the Output # (y).
# of Triangles Rule:
(x)
1
2
3
...
x
It Works!
1x +2
1x +2
1x +2
P=6
Perimeter
(y)
3
4
5
…
y
from the MCAS
Patterns in Numbers in Tables:
• Write a rule for the table below.
Input (x)
2
Output (y) 5
5
10
11
11
21
23
from the MCAS
2) Look for the Constant Difference.
Input (x)
2
5
10
11
Output (y)
5
11
21
23
•What is the change when the input #
increases by 1?
•From the 10th to the 11th the output #s
increase from 21 to 23.
So the constant difference is +2.
3) Multiply x by the Constant Difference.
Then…
4) Add or Subtract some #.
Input (x)
2
5
10
11
Output (y)
5
11
21
23
Constant Difference
2x
+1
Constant
Input #
Patterns in Numbers in Real-Life
Situations:
Write a rule for x number of rides:
from the MCAS
1) Make a Table:
In (x)
# of Rides
1
Out (y)
Cost
$
12
2
$
14
3
$
16
2) Find the Constant Difference.
In (x)
Out (y)
# of Rides Cost
$12
1
2
3
$14
$16
+$2
+$2
+$2…
So the Constant Difference is +2.
3) Multiply x by the Constant Difference.
Then…
4) Add or Subtract some #.
In (x)
Out (y)
# of Rides Cost
1
2
3
$12
$14
$16
Constant Difference
2 x +10
Input #
Constant
Patterns in Sequences of Numbers
12, 16, 20, 24…
What’s my rule?
Remember:
1) Make a Table.
2) Find the Constant Difference.
3) Multiply x by the Constant Difference.
4) Add or Subtract some #.
1) Make a Table:
(x)
1
2
3
(y)
12
16
20
+4
+4
2) Find the Constant Difference.
The Constant Difference is +4.
3) Multiply x by the Constant Difference.
Then…
4) Add or Subtract some #.
(x)
1
2
3
(y)
12
16
20
Constant Difference
4 x +8
Input #
Constant
Patterns in Linear Graphs
“Linear” means it
makes a
straight line.
Remember:
1. Make a Table.
2. Find the Constant Difference.
3. Multiply x by the Constant Difference.
4. Add or Subtract some #.
To Make a Table from a Graph:
(x)
(y)
-1
-3
0
-1
+2
+2
1
1
Find the Constant Difference.
3) Multiply x by the Constant Difference.
Then…
4) Add or Subtract some #.
(x)
-1
0
1
(y)
-3
-1
1
Constant Difference
2 x -1
Input
Constant
How to find the 10th or 100th term:
• Now that we have a rule we can
find any term we want by
evaluating for that term #.
• Just substitute the term number for
x, then simplify.
What would ‘y’ be if x = 10?
The rule for the last graph was:
2x -1
Substitute 10 for x and we get:
(2)(10) – 1 or 20 -1 = 19. So (10, 19) are
solutions for this rule,
AND (10, 19) would be a point on this
line!
What would ‘y’ be if x = 100?
2x – 1 was the rule for the graph.
Substitute 100 for x:
(2)(100) – 1 or
200 -1 = 199
So (100, 199) would be a solution for
this rule,
AND (100, 199) would be on this line!
Review
So here we have come full circle, we have:
Written algebraic expressions;
Evaluated these expressions;
Written expressions (rules) for patterns;
Evaluated these rules for specific terms.
Graphing Linear Patterns
There are 3 forms of equations
that can be graphed:
1) Slope-intercept form
2) Standard form
3) Point-slope form
Slope-Intercept Form (Slope)
• The “slope” of a line is
the measure of its
steepness.
rise
run
Or: Rise
over
Run
Y-Intercept:
• The y-intercept is the point where a
line crosses the y-axis.
-1
• Hint: Think of the word,
‘intersection’, where 2 streets
cross, in order to remember
‘intercept’.
Finding the Slope on a Graph:
The slope of the line is
rise
run.
Or: the change in y
the change in x.
Change in y = 2 2
=
Change in x = 1 1
So the slope is +2.
Kinds of Slopes:
•Slopes may be positive
(y increases as x increases);
•Slopes may be negative
(y decreases as x increases);
•Slopes may be zero
(y doesn’t change at all);
•Or Slopes may be undefined
(x doesn’t change at all).
Name the Type of Slope:
Slope-Intercept Form:
You can see both the slope and the y-intercept
slope
on the graph:
2 x -1
y-intercept
Standard Form:
• It’s easy to find the x- and y-intercept
with the standard form (Ax + By = C).
• All you need to do is substitute “0” for
x and solve for y; then substitute “0”
for y and solve for x.
Try it:
Write y = 2x -1 in standard form:
y = 2x - 1
-2x -2x
y - 2x = -1
y - (2) (0) = -1
y = -1
So the y-intercept is -1.
0 - (2) x = -1
-2
-2
x = 1/2
So the x-intercept is 0.5.
Point-Slope Form:
The point slope form (y - y1) = m(x - x1)
is easiest to use if you are given
one point and the slope of the line.
Just substitute the coordinates into the
equation. Then rewrite the equation in
slope-intercept form.
Point-Slope Form
•Suppose you did not have the graph,
but you were told that the point (2, 3) is on
the line and the slope is +2…
•You could write the equation: y - 3 = 2(x - 2),
then rewrite it in slope-intercept form.
Point-Slope Form:
You could rewrite y - 3 = 2(x - 2) to the
slope-intercept form:
y - 3 = 2(x - 2)
y - 3= 2x - 4
+3
+3
y = 2x -1
Slopes of Parallel Lines:
Two lines on the same plane
that have the same slope
will be parallel.
Slope is 0.
Slope is undefined.
Slopes of Perpendicular Lines:
Note: Perpendicular lines form right angles
at their intersection.
Two lines whose slopes are negative reciprocals
are perpendicular. The product of their slopes
will equal -1.
Are they Parallel or Perpendicular?
y = 2x + 10
y = 2x -5
y = -3x + 2
y = 1/3x + 1