Chapter 5 Notes
5-1 Relating Graphs to Events
When interpreting what a graph shows, pay attention to the labels on the axes.
Graphs can have 6 different types of basic activity:
Constant ______________
Increasing/Decreasing at a constant rate
Increasing at an Increasing Rate
Increasing at a decreasing rate
Decreasing at an Increasing Rate
Decreasing at a Decreasing Rate
5-2 Relations and Functions
A relation is a set of numbers that form a relation, such as age and shoe size. If a relation is also a function, it will assign exactly one output value for every input value. Obviously, age and shoe size is NOT a function since people of the same age will have different shoe sizes.
To tell if a relation is a function, you can plot the points (ordered pairs) and then use the vertical line test. If two points are exactly vertical, it is NOT a function. However, if no two points lie on top of each other, or vertical to each other, then the relation is considered to be a function.
When we have a function, instead of using Y =, we use the function notation, of f(x) =. This is done to show the value of the function at the value of x. So if the function is f(x) = 3x + 2, then f(6) = 3(6) + 2 = 18 + 2 = 20
Domain = all the x values or input values
Range = all the y values or output values.
5-3 Function Rules, Tables and Graphs
A Function Rule is the equation that describes the function. A Table is generally an X-Y table (also called a T-Table) showing several input and an output values for the function and of course the Graph is a visual representation of the function.
The Independent variable is graphed on the x-axis (horizontal) and the Dependent variable is graphed on the y-axis (vertical).
Given an equation:
Make a table by selecting various input values and using the equation to solve for the output values
Use the table of values to plot the ordered pairs and create the graph
Given a Table of Values:
Determine the change in the Y value (dependent value) over the change in the X Value (independent value). This is “m.” Then determine the y-value when x = 0. This is “b.” Use those values to write the function f(x) = mx + b
Use the table of values to plot the ordered pairs and create the graph
Given a Graph:
Where does the graph cross the vertical axis? This is “b.” Determine the rise over the run on the graph. This is “m.” Use those values to write the function f(x) = mx + b.
Use that equation to make a table by selecting various input values and using the equation to solve for the output values
Continuous Vs. Discrete
When graphing data, it is important to know the context of the problem. Does it make sense to have every value possible on a line? For example, if I am graphing cost vs. lbs of nuts, can I have any possible lb? Does it make sense to have ½ a lb? ¼ lb? etc? Since it does, we call this information continuous and use a solid line to represent the graph. If it doesn’t make sense to have every value, say I’m graphing cost vs. number of T-shirts purchased, where it doesn’t make sense to purchase ½ a T-shirt, then we say the information is discrete and we use a point per value line to represent the graph of the data. (Only put a point where the data actually exists and do NOT connect them!)
5-4 Writing a Function Rule
To write a function, determine first how the two values are related. Determine the rate at which the output changes in respect to the input. For example, does the out put change 5 for every change of 3 in the input? If so, then the rate is 5/3. This “rate” is the “m” in the function f(x) = mx + b. The “b” value is the “beginning” on the output, in other words, the y-value when x = 0. Can you use the table to move forward or back to determine the “b”?
The output (Y) value changes by 2 for every change of 1 in the input (X) value, so the rate, or “m” is 2/1 = 2.
Next find the value of y when x = 0.
If I went backwards on the X side to get to zero, I would go back two steps (to 1, then to 0) so I also have to go back two steps on the Y side, keeping in mind each step there is 2 units (first down to 6, then to 4) so the “b” = 4.
Use those values to write the equation y = mx + b or the function f(x) = mx + b
Y = 2x + 4 or f(x) = 2x + 4
If the b value (y-intercept) is zero, the graph will pass through the origin.
5-5 Direct Variation
An equation in the form y = kx is a direct variation. K is essentially the same thing as “m”, and b=0. So if you have a line that passes through the origin, it is most likely a direct variation – one item (the y) varies (or changes) directly with the other (the x value).
The “k” is called the constant of variation. If you were to solve for k in the equation y = kx, you discover that k = y/x . Given an ordered pair then, you can solve for a constant of variation and write the direct variation equation.
Example: Write the direct variation equation through the ordered pair (6, 8).
Well, k = y/x so k = 8/6 = 4/3. Thus the equation is y = 4/3 x Simple as that.
If an equation in slope intercept form (y = mx + b) has a b-value, it can NOT be a
direct variation, only if the b-value is 0.
5-6 Inverse Variation
An equation in the form y = k/x is an Inverse Variation. (As long as k≠0) When you solve for “k” in this case, you find that k = xy. The graph of an Inverse Variation is a curve.
Example: Write the equation for the Inverse Variation of a graph that passes through (6, 8)
So k = xy = 6*8 = 48. Substitute into the equation y = k/x and you get y = 48/x
Example: Given (3, 8) & (6, y) Find y if the two points are on the graph of an inverse variation.
Use the first point to find k. k = xy = 3*8 = 24. Plug that into the equation y = k/x to get y = 24/x then substitute in the x value from the other point.
Y = 24/6 = 4 so the missing y is y = 4.
5-7 Describing Number Patterns
Number patterns can be difficult to determine so often we use Inductive Reasoning to make conclusions about what will happen next. We can’t prove it, but based on what we have seen in the past, we can justify, with a fair amount of confidence, that we know what will happen. The conclusion we make using this type of reasoning is called a Conjecture.
Number patterns are often called sequences. There are two kinds of sequences that we often see. One is called an Arithmetic Sequence where the next term increases or decreases by the same non-zero number each time. This change is called a common difference.
The other is a Geometric Sequences where the next term is determined by multiplying or dividing by the same non-zero number each time. Here the change is called the Common Ratio.
Arithmetic Sequences use addition and Subtraction
Geometric Sequences use multiplication and division
If you presume an arithmetic sequence, you can determine any term by using the following Rule:
Nth term = first term + (term you want – 1) times the common difference
A(n) = ao + (n – 1) d
Example: Find the 15th term of the sequence { 3, 5, 7, 9…} The common difference is 2 so:
A(15) = 3 + (15 -1) 2
= 3 + 14*2
= 3 + 28
= 31
This is certainly faster than figuring it out, especially when looking for very large
When interpreting what a graph shows, pay attention to the labels on the axes.
Graphs can have 6 different types of basic activity:
Constant ______________
Increasing/Decreasing at a constant rate
Increasing at an Increasing Rate
Increasing at a decreasing rate
Decreasing at an Increasing Rate
Decreasing at a Decreasing Rate
5-2 Relations and Functions
A relation is a set of numbers that form a relation, such as age and shoe size. If a relation is also a function, it will assign exactly one output value for every input value. Obviously, age and shoe size is NOT a function since people of the same age will have different shoe sizes.
To tell if a relation is a function, you can plot the points (ordered pairs) and then use the vertical line test. If two points are exactly vertical, it is NOT a function. However, if no two points lie on top of each other, or vertical to each other, then the relation is considered to be a function.
When we have a function, instead of using Y =, we use the function notation, of f(x) =. This is done to show the value of the function at the value of x. So if the function is f(x) = 3x + 2, then f(6) = 3(6) + 2 = 18 + 2 = 20
Domain = all the x values or input values
Range = all the y values or output values.
5-3 Function Rules, Tables and Graphs
A Function Rule is the equation that describes the function. A Table is generally an X-Y table (also called a T-Table) showing several input and an output values for the function and of course the Graph is a visual representation of the function.
The Independent variable is graphed on the x-axis (horizontal) and the Dependent variable is graphed on the y-axis (vertical).
Given an equation:
Make a table by selecting various input values and using the equation to solve for the output values
Use the table of values to plot the ordered pairs and create the graph
Given a Table of Values:
Determine the change in the Y value (dependent value) over the change in the X Value (independent value). This is “m.” Then determine the y-value when x = 0. This is “b.” Use those values to write the function f(x) = mx + b
Use the table of values to plot the ordered pairs and create the graph
Given a Graph:
Where does the graph cross the vertical axis? This is “b.” Determine the rise over the run on the graph. This is “m.” Use those values to write the function f(x) = mx + b.
Use that equation to make a table by selecting various input values and using the equation to solve for the output values
Continuous Vs. Discrete
When graphing data, it is important to know the context of the problem. Does it make sense to have every value possible on a line? For example, if I am graphing cost vs. lbs of nuts, can I have any possible lb? Does it make sense to have ½ a lb? ¼ lb? etc? Since it does, we call this information continuous and use a solid line to represent the graph. If it doesn’t make sense to have every value, say I’m graphing cost vs. number of T-shirts purchased, where it doesn’t make sense to purchase ½ a T-shirt, then we say the information is discrete and we use a point per value line to represent the graph of the data. (Only put a point where the data actually exists and do NOT connect them!)
5-4 Writing a Function Rule
To write a function, determine first how the two values are related. Determine the rate at which the output changes in respect to the input. For example, does the out put change 5 for every change of 3 in the input? If so, then the rate is 5/3. This “rate” is the “m” in the function f(x) = mx + b. The “b” value is the “beginning” on the output, in other words, the y-value when x = 0. Can you use the table to move forward or back to determine the “b”?
The output (Y) value changes by 2 for every change of 1 in the input (X) value, so the rate, or “m” is 2/1 = 2.
Next find the value of y when x = 0.
If I went backwards on the X side to get to zero, I would go back two steps (to 1, then to 0) so I also have to go back two steps on the Y side, keeping in mind each step there is 2 units (first down to 6, then to 4) so the “b” = 4.
Use those values to write the equation y = mx + b or the function f(x) = mx + b
Y = 2x + 4 or f(x) = 2x + 4
If the b value (y-intercept) is zero, the graph will pass through the origin.
5-5 Direct Variation
An equation in the form y = kx is a direct variation. K is essentially the same thing as “m”, and b=0. So if you have a line that passes through the origin, it is most likely a direct variation – one item (the y) varies (or changes) directly with the other (the x value).
The “k” is called the constant of variation. If you were to solve for k in the equation y = kx, you discover that k = y/x . Given an ordered pair then, you can solve for a constant of variation and write the direct variation equation.
Example: Write the direct variation equation through the ordered pair (6, 8).
Well, k = y/x so k = 8/6 = 4/3. Thus the equation is y = 4/3 x Simple as that.
If an equation in slope intercept form (y = mx + b) has a b-value, it can NOT be a
direct variation, only if the b-value is 0.
5-6 Inverse Variation
An equation in the form y = k/x is an Inverse Variation. (As long as k≠0) When you solve for “k” in this case, you find that k = xy. The graph of an Inverse Variation is a curve.
Example: Write the equation for the Inverse Variation of a graph that passes through (6, 8)
So k = xy = 6*8 = 48. Substitute into the equation y = k/x and you get y = 48/x
Example: Given (3, 8) & (6, y) Find y if the two points are on the graph of an inverse variation.
Use the first point to find k. k = xy = 3*8 = 24. Plug that into the equation y = k/x to get y = 24/x then substitute in the x value from the other point.
Y = 24/6 = 4 so the missing y is y = 4.
5-7 Describing Number Patterns
Number patterns can be difficult to determine so often we use Inductive Reasoning to make conclusions about what will happen next. We can’t prove it, but based on what we have seen in the past, we can justify, with a fair amount of confidence, that we know what will happen. The conclusion we make using this type of reasoning is called a Conjecture.
Number patterns are often called sequences. There are two kinds of sequences that we often see. One is called an Arithmetic Sequence where the next term increases or decreases by the same non-zero number each time. This change is called a common difference.
The other is a Geometric Sequences where the next term is determined by multiplying or dividing by the same non-zero number each time. Here the change is called the Common Ratio.
Arithmetic Sequences use addition and Subtraction
Geometric Sequences use multiplication and division
If you presume an arithmetic sequence, you can determine any term by using the following Rule:
Nth term = first term + (term you want – 1) times the common difference
A(n) = ao + (n – 1) d
Example: Find the 15th term of the sequence { 3, 5, 7, 9…} The common difference is 2 so:
A(15) = 3 + (15 -1) 2
= 3 + 14*2
= 3 + 28
= 31
This is certainly faster than figuring it out, especially when looking for very large