Chapter 6 Notes
6-1 Rate of Change and Slope
Rate of Change allows you to see the relationship between two quantities that are changing. If one quantity is dependent upon the other (example: miles traveled depends on time traveled) then the rate of change is the change in the dependent variable divided by the change in the independent variable.
When we look at a graph, the dependent variable is typically graphed on the y-axis, and the independent on the x-axis. In short then, the rate of change is the change in the y value over the change in the x value.
We can find the rate if change either by looking at a table of values or by looking at a graph.
In a table, we look for the change in y over the change in x.
On a graph, we look for the rise over the run - which is the change in y over the change in x since y values are up and down, (rise) and x values are right and left, (run)
The bigger the change, the steeper the slope of the line. Graphically, the rate of change IS the slope.
The slope (m) then can be determined by finding the change in two y values, over the change in two x values.
Slope (m) = y2 – y1/x2 – x1
A positive slope should have an upward trend (or point up from left to right) and a negative slope should have a downward trend or point down from left to right).
A line with zero slope would be a line whose slope equation had a zero on top. That means that the values of y didn’t change. The only line whose y-values do not change is a horizontal line.
A vertical line has no change in the x values (since the line never moves left or right) – this creates a value of zero on the bottom of the fraction when we figure slope. Since we can’t divide by zero, we say that a vertical line has undefined slope.
6-2 Slope-Intercept Form
Every form of Algebraic Equation has what is called a “Parent Function” – this is the basic form of that function. The parent function for a Linear equation (which forms a straight line) is y = x. This is similar to the direct variation we looked at in chapter 5. It passes through the origin, and the parent function has a slope of one.
Once you have established the parent function, then we start changing the slope and y-intercept to create other linear functions.
The slope (m) changes how steep the line is, while the y-intercept (b) changes where the line crosses the y-axis. Think of it as a shift up or down from the origin.
We can write equations in Slope Intercept form quite easily from a graph, (just determine the slope and the intercept) and we can also graph an equation in Slope Intercept form quite easily as well (start at the intercept, make a point, then follow the slope, rise over run.)
y=mx + b
Example:
Graph y = 4x + 6
6-3 Applying Linear Functions
When given a “real life” situation for which you need to write a linear function, begin by determining what the two variables are, and which one is dependent and which one is independent.
Then use the information provided to get ordered pairs from which you can determine the rate of change, or the slope.
To get the y-intercept (b) determine what the dependent value would be if the independent value were zero.
Once you have your slope and your y-intercept, you can write your function in the form f(x) = mx + b
Example:
Suppose you go to a theme park. It costs you $3 to get in, and then another $2 for every ride you go on. Your total cost will depend on the number of rides you go on, so cost depends on rides. The change in cost per ride is 2, so the slope (m) is 2. If you were to go on zero rides, it would still cost you $3 to get in, so the y-intercept (b) is 3. Use these to write the cost function f(x) = 2x + 3
6-4 Standard Form
Slope Intercept form is not the only form for writing linear equations, although it is probably the easiest for graphing.
Standard Form is also very useful, and honestly the most common in upper level mathematics.
Standard Form looks like Ax + By = C, where A, B, and C are real numbers, and both A and B can NOT be zero.
In standard form, you can easily get your y-intercept by allowing x=0. Likewise then you can also get your x-intercept (where the graph crosses the x-axis) by allowing y=0. Once you know both of these intercepts, you can also easily graph from Standard Form.
Standard form for a Horizontal line is y = b (since m would be zero in y=mx + b)
And Standard Form for a Vertical Line is x = c
To find the slope in Standard Form, you can rearrange the standard from equation into slope intercept form and discover the slope is –A/B
6-5 Point Slope Form and Writing Linear Equations
Another form for linear equations is called Point-Slope Form. This form is useful when you are given a point, and a slope. (Duh!)
If your point is (x1, y1) then the form looks like y – y1 = m(x – x1)
If you are given the equation to graph, identify the point, using it as a starting point, follow the slope.
Example: Graph y – 5 = ½ (x + 3)
Identify the point: (-3, 5) – Notice the x coordinate is negative, because the equation called for subtraction, but it was positive in the given equation. Two negatives make the positive, so the x-coordinate must be negative)
Now starting at the point (-3, 5) on your graph, go up 1 and over 2 (follow the slope of ½ ) to get the next point. Connect the dots.
If you are given two points, use them to find your slope, then select one to use in the Point-Slope Form equation. It does not matter which one you use – when converted to Standard Form, either one will give the same result!
6-6 Parallel and Perpendicular Lines
Parallel Lines are lines that lie in the same plane, but never cross. Think of them as straight railroad tracks. If you saw railroad tracks on a map, you would notice that both lines have the same steepness, otherwise the wheel would not always be the same distance apart, and the train would derail. Since both lines have the same steepness, we say they have the same slope.
Parallel Lines have the same slope
If you want to find a line parallel to a given line through a point, use the slope from the first line, and the given point, using Point-Slope form and you can get your equation.
Example: Find the equation of the line that passes through (2, -5) and is parallel to y = ¾ x + 7 in Slope Intercept Form.
Use the slope of ¾ and the point (2, -5) in Point-Slope Form, then convert to Slope Intercept Form:
y – (-5) = ¾ (x – 2)
y + 5 = ¾ x – 6/4
y = ¾ x – 6/4 – 5
y = ¾ x – 26/4
y = ¾ x – 13/2 Notice, the equation is the same as the given equation except for the y-intercept.
To determine then if lines are parallel, simply find the slope of each – if they are the same, they are parallel.
Perpendicular Lines form a right angle. Since they go in opposite directions, one line should have a positive slope while the other has a negative slope. The product of the slopes of perpendicular lines is (-1), thus the slopes themselves must be negative reciprocals.
If one line has a slope of 5/8 then the perpendicular line would have a slope of -8/5
6-7 Scatter Plots and Equations of Lines
Recall: A scatter plot of a graph showing the relation for two different sets of data. (Such as age and shoe size for several different people, where each point represents one person.)
A scatter plot has either positive, negative, or no trend. If the trend is positive or negative, we can estimate a line of best fit by drawing a line to approximate the data, then picking two points on the line and writing the equation.
We can also do this using a graphing calculator. The calculator will provide us with a correlation coefficient “r” which tells how close the line is to the actual data. The closer r is to 1, the better the fit.
To use your calculator, make sure before you begin that your diagnostics are turned ON.
To do this, Hit 2nd , Catalog (0), then scroll down to DiagnosticOn, Enter, Enter again. You should only have to do this once.
To enter data for a scatter plot:
STAT
#1
This will take you to your lists. If there is something in your list that you wish to clear, arrow up until the list number is highlighted (such as L2) and then hit Clear, Enter.
You may now begin entering your data points.
When you are done and ready to graph:
Turn on your stat plot: 2nd , Stat Plot (Y=), Enter, Enter on the ON, arrow down and enter on the first option, arrow down and make sure the XList and YList are the correct Lists that you used. Select your mark, Hit GRAPH, and ZOOM 9.
To get the line of best fit:
STAT > CALC
#4
You should see “LinReg(ax+b)” on your home screen. Enter L1, L2 by using 2nd #1, comma (above the 7), and 2nd #2.
You should now see displayed
a =
b=
r2=
r=
a is your slope (m). You can use these values to write your equation.
6-8 Graphing Absolute Value Equations
The Graph of an Absolute Value equation is easy to graph if you know your parent function and rules for shifts.
The Parent Function of an Absolute Value Y = |x| is a V that meets at the origin with a slope of + 1.
Shifts are either on y (also known as your y intercept – they go in the same direction, up for positive, down for negative) or on x, in which case they are contained with the x in the absolute value, and they shift in the opposite direction.
Example:
Y = |x| + 2 would be a V that meets on the Y-Axis at +2
Y = |x + 2| would be a V that meets on the X-Axis at -2.
These shifts are called Translations
Rate of Change allows you to see the relationship between two quantities that are changing. If one quantity is dependent upon the other (example: miles traveled depends on time traveled) then the rate of change is the change in the dependent variable divided by the change in the independent variable.
When we look at a graph, the dependent variable is typically graphed on the y-axis, and the independent on the x-axis. In short then, the rate of change is the change in the y value over the change in the x value.
We can find the rate if change either by looking at a table of values or by looking at a graph.
In a table, we look for the change in y over the change in x.
On a graph, we look for the rise over the run - which is the change in y over the change in x since y values are up and down, (rise) and x values are right and left, (run)
The bigger the change, the steeper the slope of the line. Graphically, the rate of change IS the slope.
The slope (m) then can be determined by finding the change in two y values, over the change in two x values.
Slope (m) = y2 – y1/x2 – x1
A positive slope should have an upward trend (or point up from left to right) and a negative slope should have a downward trend or point down from left to right).
A line with zero slope would be a line whose slope equation had a zero on top. That means that the values of y didn’t change. The only line whose y-values do not change is a horizontal line.
A vertical line has no change in the x values (since the line never moves left or right) – this creates a value of zero on the bottom of the fraction when we figure slope. Since we can’t divide by zero, we say that a vertical line has undefined slope.
6-2 Slope-Intercept Form
Every form of Algebraic Equation has what is called a “Parent Function” – this is the basic form of that function. The parent function for a Linear equation (which forms a straight line) is y = x. This is similar to the direct variation we looked at in chapter 5. It passes through the origin, and the parent function has a slope of one.
Once you have established the parent function, then we start changing the slope and y-intercept to create other linear functions.
The slope (m) changes how steep the line is, while the y-intercept (b) changes where the line crosses the y-axis. Think of it as a shift up or down from the origin.
We can write equations in Slope Intercept form quite easily from a graph, (just determine the slope and the intercept) and we can also graph an equation in Slope Intercept form quite easily as well (start at the intercept, make a point, then follow the slope, rise over run.)
y=mx + b
Example:
Graph y = 4x + 6
6-3 Applying Linear Functions
When given a “real life” situation for which you need to write a linear function, begin by determining what the two variables are, and which one is dependent and which one is independent.
Then use the information provided to get ordered pairs from which you can determine the rate of change, or the slope.
To get the y-intercept (b) determine what the dependent value would be if the independent value were zero.
Once you have your slope and your y-intercept, you can write your function in the form f(x) = mx + b
Example:
Suppose you go to a theme park. It costs you $3 to get in, and then another $2 for every ride you go on. Your total cost will depend on the number of rides you go on, so cost depends on rides. The change in cost per ride is 2, so the slope (m) is 2. If you were to go on zero rides, it would still cost you $3 to get in, so the y-intercept (b) is 3. Use these to write the cost function f(x) = 2x + 3
6-4 Standard Form
Slope Intercept form is not the only form for writing linear equations, although it is probably the easiest for graphing.
Standard Form is also very useful, and honestly the most common in upper level mathematics.
Standard Form looks like Ax + By = C, where A, B, and C are real numbers, and both A and B can NOT be zero.
In standard form, you can easily get your y-intercept by allowing x=0. Likewise then you can also get your x-intercept (where the graph crosses the x-axis) by allowing y=0. Once you know both of these intercepts, you can also easily graph from Standard Form.
Standard form for a Horizontal line is y = b (since m would be zero in y=mx + b)
And Standard Form for a Vertical Line is x = c
To find the slope in Standard Form, you can rearrange the standard from equation into slope intercept form and discover the slope is –A/B
6-5 Point Slope Form and Writing Linear Equations
Another form for linear equations is called Point-Slope Form. This form is useful when you are given a point, and a slope. (Duh!)
If your point is (x1, y1) then the form looks like y – y1 = m(x – x1)
If you are given the equation to graph, identify the point, using it as a starting point, follow the slope.
Example: Graph y – 5 = ½ (x + 3)
Identify the point: (-3, 5) – Notice the x coordinate is negative, because the equation called for subtraction, but it was positive in the given equation. Two negatives make the positive, so the x-coordinate must be negative)
Now starting at the point (-3, 5) on your graph, go up 1 and over 2 (follow the slope of ½ ) to get the next point. Connect the dots.
If you are given two points, use them to find your slope, then select one to use in the Point-Slope Form equation. It does not matter which one you use – when converted to Standard Form, either one will give the same result!
6-6 Parallel and Perpendicular Lines
Parallel Lines are lines that lie in the same plane, but never cross. Think of them as straight railroad tracks. If you saw railroad tracks on a map, you would notice that both lines have the same steepness, otherwise the wheel would not always be the same distance apart, and the train would derail. Since both lines have the same steepness, we say they have the same slope.
Parallel Lines have the same slope
If you want to find a line parallel to a given line through a point, use the slope from the first line, and the given point, using Point-Slope form and you can get your equation.
Example: Find the equation of the line that passes through (2, -5) and is parallel to y = ¾ x + 7 in Slope Intercept Form.
Use the slope of ¾ and the point (2, -5) in Point-Slope Form, then convert to Slope Intercept Form:
y – (-5) = ¾ (x – 2)
y + 5 = ¾ x – 6/4
y = ¾ x – 6/4 – 5
y = ¾ x – 26/4
y = ¾ x – 13/2 Notice, the equation is the same as the given equation except for the y-intercept.
To determine then if lines are parallel, simply find the slope of each – if they are the same, they are parallel.
Perpendicular Lines form a right angle. Since they go in opposite directions, one line should have a positive slope while the other has a negative slope. The product of the slopes of perpendicular lines is (-1), thus the slopes themselves must be negative reciprocals.
If one line has a slope of 5/8 then the perpendicular line would have a slope of -8/5
6-7 Scatter Plots and Equations of Lines
Recall: A scatter plot of a graph showing the relation for two different sets of data. (Such as age and shoe size for several different people, where each point represents one person.)
A scatter plot has either positive, negative, or no trend. If the trend is positive or negative, we can estimate a line of best fit by drawing a line to approximate the data, then picking two points on the line and writing the equation.
We can also do this using a graphing calculator. The calculator will provide us with a correlation coefficient “r” which tells how close the line is to the actual data. The closer r is to 1, the better the fit.
To use your calculator, make sure before you begin that your diagnostics are turned ON.
To do this, Hit 2nd , Catalog (0), then scroll down to DiagnosticOn, Enter, Enter again. You should only have to do this once.
To enter data for a scatter plot:
STAT
#1
This will take you to your lists. If there is something in your list that you wish to clear, arrow up until the list number is highlighted (such as L2) and then hit Clear, Enter.
You may now begin entering your data points.
When you are done and ready to graph:
Turn on your stat plot: 2nd , Stat Plot (Y=), Enter, Enter on the ON, arrow down and enter on the first option, arrow down and make sure the XList and YList are the correct Lists that you used. Select your mark, Hit GRAPH, and ZOOM 9.
To get the line of best fit:
STAT > CALC
#4
You should see “LinReg(ax+b)” on your home screen. Enter L1, L2 by using 2nd #1, comma (above the 7), and 2nd #2.
You should now see displayed
a =
b=
r2=
r=
a is your slope (m). You can use these values to write your equation.
6-8 Graphing Absolute Value Equations
The Graph of an Absolute Value equation is easy to graph if you know your parent function and rules for shifts.
The Parent Function of an Absolute Value Y = |x| is a V that meets at the origin with a slope of + 1.
Shifts are either on y (also known as your y intercept – they go in the same direction, up for positive, down for negative) or on x, in which case they are contained with the x in the absolute value, and they shift in the opposite direction.
Example:
Y = |x| + 2 would be a V that meets on the Y-Axis at +2
Y = |x + 2| would be a V that meets on the X-Axis at -2.
These shifts are called Translations