4.2 Patterns and Linear Functions: Independent Variable: The variable that is not dependent....
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Transcript of 4.2 Patterns and Linear Functions: Independent Variable: The variable that is not dependent....
4.2 Patterns and Linear Functions:
Independent Variable: The variable that is not dependent.
Dependent Variable: The variable that depends upon the value of another.
Input: The values of the independent variable.
Output: The values of the dependent variable.
Linear Function: A function whose graph is a nonvertical line or part of a nonvertical line.
Function f(x): A relationship that pairs one input to exactly one output (x, y)
In math we use tables, words, equations, set of ordered pairs and graphs to represent a relationship between two variables.
1. Geometric Relationships,
2. Linear Functions
This can be done when we are presented with the following:
GEOMETRIC RELATIONSHIPS:
Ex: Use words, an equation, a table and a graph to represent the relationship between the number of rectangles and the perimeter of the figure.
Words:
Look at the figure, multiply the number of rectangles by 2 to get the total lengths of the top an bottom sides of the combined figure.Then add 2 times the length of the left and right sides of the combined figure to obtain the final answer for the total perimeter of the figure.
Equation:Look at the figure: We must realize that the only thing that is changing is the number of the short side (width). Also, the number of the length is constant which is = 12.
Using this info we see that the equation is:
y or f(x) = twice the number of short + twice the length.
f(x) = 2x + 12
Table:Looking at the figures we see that:X = number of rectangles (independent)y = Perimeter of the figure (depends on figure)
Number of rectangles (x)
Perimeter (y)y = 2l + 2w
Ordered Pairs(x, y)
1 2(6) + 2(1) = 14 (1, 14)
2 2(6) + 2(2) = 16 (2, 16)
3 2(6) + 2(3) = 18 (3, 18)
4 2(6) + 2(4) = 20 (4, 20)
5 2(6) + 2(5) = 22 (5, 22)
Graph:Pe
rimet
er
Figure
Ordered Pairs(x, y)
(1, 14)
(2, 16)
(3, 18)
(4, 20)
(5, 22)
24 68
10 12 14 16 18 20 22
1 2 3 4 5
YOU TRY IT:
Use one method to represent the relationship between the number of triangles and the perimeter.
1
1
1 1
1
1
11
1
1
1 1
Words:Triangles = 1 Perimeter = 3
Look at the figure, The perimeter is 2 more than the number of triangles.
1
1
1
1
1
1
1 Triangles = 2 Perimeter = 4
1
1
1
11
Triangles = 3 Perimeter = 5
Equation:Again, the perimeter [ y or f(x)] is 2 more than the number of triangles (x)
y = x + 2
f(x) = x + 2
Table:Looking at the figures we see that:X = number of rectangles (independent)y = Perimeter of the figure (depends on figure)
Number of rectangles (x)
Perimeter (y)y = x + 2
Ordered Pairs(x, y)
1 1 + 2 = 3 (1, 3)
2 2 + 2 = 4 (2, 4)
3 3 + 2 = 5 (3, 5)
4 4 + 2 = 6 (4, 6)
5 5 + 2 = 7 (5, 7)
Graph:Pe
rimet
er
Figure
Ordered Pairs(x, y)
(1, 3)
(2, 4)
(3, 5)
(4, 6)
(5, 7)
12 34
5 6 7 8 9 10
1 2 3 4 5
Q: What is the value of y if x = 0?
LINEAR FUNCTIONS: Data from a table can be scrutinize to see if it is a linear relation. In order for us to make the final decision, we first must see how the y – function, changes for each x in the table.
Ex:Is there a linear relation in this table?
Number of Photos (x)
0 1 2 3
Memory (y)
512 509 506 503
To answer the question we must take a look at what is happening in the table.
Number of Photos (x)
0 1 2 3
Memory (y)
512 509 506 503
+ 1 + 1 + 1
- 3 - 3 - 3
The dependent variable y decreases by 3The independent variable x increases by 1The starting memory is 512 MG
Taking the info to consideration, we can see that the equation for the problem is:
Number of Photos (x)
0 1 2 3
Memory (y)
512 509 506 503
+ 1 + 1 + 1
- 3 - 3 - 3
The dependent variable y decreases by 3
y = 512 – 3x
YOU TRY IT:
For the table, determine whether the relationship is a linear function. Then represent the relationship using words, an equation and a graph.
Hours (x) Money(y)
0 10
1 18
2 26
3 34
YOU TRY IT: (SOLUTION)
Looking at both variables, we have:Hours (x) Money(y)
0 10
1 18
2 26
3 34
+8
+8
+8
+1
+1
+1
Both, the x and y are changing at a constant rate.
YOU TRY IT: (Words Solution):
A person had 10 dollars and then starts a job where he earns eight dollars per hour.
YOU TRY IT: (Equation Solution):
We started with 10 dollars and earn 8 after each hour.
y = 8x + 10
f(x) = 8x + 10
YOU TRY IT: (Graph Solution)M
oney
$
Hours
Hours(x)
Dollars(y)
0 10
1 18
2 26
3 34
4 ?
Q: What will the total money after 4 hrs?.
5
10
15
20
25
30
35
1 2 3
VIDEOS: Linear Functions
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/analyzing-functions-algebra/v/constructing-and-interpreting-a-linear-function
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/analyzing-functions-algebra/v/constructing-a-linear-function-word-problem