expressions, rules, and graphing linear equations by lauren mccluskey
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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:. - PowerPoint PPT PresentationTRANSCRIPT
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 2 m + 8
• 3x + (-15) -2x + 5 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 m = 2 2 + 8 = 10
• r – 3 r = 5 5 – 3 = 2
“Algebra I” by M. Yuskaitis
Translating Words to Algebraic Expressions• Sum Difference
• More than Less than
• Plus Minus
• Increased Decreased
• Altogether
“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 + 10n - 4
x - 6n + 8n + 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 may be seen in:
• Geometric Figures
• Numbers in Tables
• Numbers in Real-life Situations
• Sequences of Numbers
• Linear Graphs
Patterns are predictable.
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.
from the MCAS
P=3
P= 4
P=5
P=6
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.
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 ? 3 2 4 3 5 ... … x y
from the MCAS
P = 3
P = 4
P = 5
P = 6
2)Find the Constant Difference: How did the output change?
Perimeter (y)
3 4 5 6 … p from the MCAS
+1
+1
+1
P = 3
P = 4
P = 5
P = 6
3) Multiply by the Input # (x). 4) Then Add or Subtract some # to get the Output # (y).
# of Triangles Rule: Perimeter
(x) (y) 1 1x +2 3 2 1x +2 4 3 1x +2 5 ... … x y
It Works! from the MCAS
P = 3
P = 4
P = 5
P = 6
Patterns in Numbers in Tables:
• Write a rule for the table below.
Input (x) 2 5 10 11
Output (y) 5 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
2Constant Difference
Input #
Constant
+1x
Patterns in Numbers in Real-Life
Situations:
from the MCAS
Write a rule for x number of rides:
1) Make a Table:
In (x)
# of Rides
Out (y)
Cost
1 $
2 $
3 $
12
14
16
2) Find the Constant Difference.
In (x)
# of Rides
Out (y)
Cost
1 $12
2 $14
3 $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)
# of Rides
Out (y)
Cost
1 $12
2 $14
3 $16
Constant Difference
Input # Constant
2 x +10
Patterns in Sequences of Numbers
Remember:
1) Make a Table.
2) Find the Constant Difference.
3) Multiply x by the Constant Difference.
4) Add or Subtract some #.
12, 16, 20, 24…
What’s my rule?
1) Make a Table:
(x) (y)
1 12
2 16
3 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) (y)
1 12
2 16
3 20
Constant Difference
Input # Constant
4 x +8
Patterns in Linear Graphs
Remember: 1. Make a Table.2. Find the Constant Difference. 3. Multiply x by the Constant Difference.4. Add or Subtract some #.
“Linear” means it makes a straight line.
To Make a Table from a Graph: (x) (y)
-1 -3
0 -1
1 1
Find the Constant Difference.
+2
+2
3) Multiply x by the Constant Difference. Then…4) Add or Subtract some #.
(x) (y)
-1 -3
0 -1
1 1
Constant Difference
Constant
2 x -1
Input
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!
ReviewSo 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 overRun
Y-Intercept:
• The y-intercept is the point where a line crosses the y-axis.
• Hint: Think of the word, ‘intersection’, where 2 streets cross, in order to remember ‘intercept’.
-1
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 2Change 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:
slope
y-intercept
2 x -1
You can see both the slope and the y-intercept on the graph:
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 -2xy - 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 +3y = 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:
Two lines whose slopes are negative reciprocals are perpendicular. The product of their slopes will equal -1.
Note: Perpendicular lines form right angles at their intersection.
Are they Parallel or Perpendicular?
y = 2x + 10 y = 2x -5
y = -3x + 2y = 1/3x + 1