If a car is driven at 50 miles per hour, the distance it travels can be represented by the equation: distance = 50 multiplied by time or d=50t, where d is distance in miles after t hours. The variable t represents time in hours and can represent any number of hours traveled. The variable d represents total distance traveled during t hours.
This constant rate of change can be observed as a pattern in a table. As t (time) increases from 0 to 1 hours, d (distance) increases from 0 to 50 miles traveled. As time continues to pass, and the car continues to travel, t increases from 1 to 2 hours, and d increases another 50 miles, for a total of 100 miles traveled. This constant rate of change, or pattern of change, continues as long as the car travels at the steady rate of 50 miles per hour.
The table below represents the data which would be obtained using the equation (rule) d=50t
Can you determine the missing d (distance) intervals? Look for the pattern of change from one interval to the next, and continue the pattern for the 5, 6, 7 and 8 hour intervals.
How did you do?
If you graphed the data in the table above, it would look like the graph below. The straight line indicates the constant rate of change of 50 from one interval to the next. This is a linear relationship.
In the above graph, the constant rate of change shows up as:
b. Create a table for the above equation.
c. Now sketch, or with a graphing calculator graph this information and compare and contrast it with the graph of the equation in Example 1.
d. What would a graph of the equations in Example 1 and Challenge 1 look like on the same set of axes?
Does your graph look like this one?
e. What is the y-intercept of the lines in the graph of both equations?
To really understand linear relationships, you must understand what nonlinear relationships are and how to recognize them: A nonlinear relationship happens when there is no constant rate of change in the y variable.
From the three tables below, can you discover which one does not represent a linear relationship?
What would the equations for each of the tables in Example 2 be?
Answer 1, 2, or 3 using the table above:
Are you getting it?
If we investigate equations of the form y=Mx, where m represents slope. (If you do not understand this, review the overview at the top of the page.) We can discover several patterns. Let's experiment with y=Mx by substituting different values for m.
Let's use the values (-4, -2, 0, 2, 4) for m with the x values from the table below to solve for y. Start with -4 for m and -1 for x and check your answers as you solve for each missing y value. You will have five y values for each x value--that's one for each m value.
To do this use the equation format, y=Mx and substitute the m and x values from above. When you do this, you will solve the first example like this: y=-4 multiplied by -1, which results in y=4 as the solution.
As you solve for the remaining y values, check your answers in the chart below. Be sure to select the y= answers as you check them, this will complete the table as you complete your work, and allow you to think about relationships between the equations used to create it.
Think about what the graphs of these equations would look like, or experiment with a graphing calculator. If you do not have a graphing calculator, you can use an online version by clicking the link at the end of this paragraph. Once you have accessed the online graphing calculator link, be sure to read the instructions before you proceed to the equation graphing calculator option. Don't forget to come back and finish working through this example.
What do you think would happen to the y values in the above chart if you added +b to the equation y=Mx? Use b=5 and explore the change in y in the chart below.
Suppose you are planning a group bike tour, and the cost to rent bikes is $150 plus $10 per bike. Using the general equation format, y=Mx+b, we can represent this situation with the equation: C=10n+150: where C takes the place of y and represents the cost in dollars; n takes the place of x and represents the number of bikes; $150 takes the place of b; and $10 takes the place of m.
Since there is a constant rate being charged for each bike ($10), there will be a linear pattern in the relationship between the total cost and the number of bikes. This constant rate of change can be observed in the table below.
Can you determine the missing C (cost) values? Look for the pattern of change from one interval to the next, and continue the pattern for renting 5, 6, 7 and 8 bikes.
If you graphed the data in the table above, it would look like the graph below.
Notice that as n (number of bikes) increases from 0 to 1, C (cost) increases by 10; as n increases from 1 to 2, C increases by 10 again. Therefore, 10 is the constant rate of change. The graph is a straight line illustrating a linear relationship between the number of bikes rented and the total cost to rent those bikes.
What would the graph of the equations in Example 4 and in this challenge look like if the were graphed together? Click to see graph.
What would happen if you manipulate the general format equation y=Mx+b and change it to y=x+b? Notice that you are now working with an equation that does not allow you to alter the slope of the line by changing the m value. All you are left with is x, y, and b, which is the y-intercept.
Here we go: Let's see what happens when we use the equation y=x+b and substitute different values for b. Remember, b is the variable used to symbolize y-intercept. Solve these equations:
What is the slope of each of these equations?
Symbolic expressions, tables, and graphs can be helpful in solving equations. Which method you choose depends upon what you already know, what you are comfortable using, or are being asked to use. For example, in Example 4 above, you might want to know the cost to rent 75 bikes or to know how many bikes you can rent for $300. In either case, you are solving for one of the variables: n (number of bikes), or C (Cost).
You could find the cost to rent 75 bikes, and how many bikes you could rent for $300 by reading the table or the graph, or by using the equation in Example 4.
Make Sure you understand the connection between an equation, a table, and a graph.
Using the equation in Challenge 4 (C=10n+150):
In the general format equation, y=Mx+b, the variable m represents slope.
You have discovered that the slope of a line is directly related to the constant rate of change in a linear relationship. You can find the constant rate of change in various ways.
a. Constant rate of change can be determined from a verbal situation, which results in an equation, like in Examples 1 and 4 where the speed of the car and the cost to rent a bike each represent constant rate of change. In Example 1, the constant rate of change is 50. In Example 2, the constant rate of change is 10.
b. Constant rate of change can be read from a table by noting the difference between each interval like in Table 1 and Table 2 of Example 2. The constant rate of change in Table 1 is 0. The constant rate of change in Table 2 is -3.
c. Constant rate of change can be determined from a graph by finding the ratio of vertical change to horizontal change of a line. This can be done by calculating the difference between two points on a line.
In the graph below, the points (1,4) and (3,10) lie on a line.
Find other points on the line and test the equation by substituting x or y and solving for the other variable.
In the general format equation, y=Mx+b, the variable b represents y-intercept.
You can find the y-intercept of an equation in various ways:
a. Y-intercept can be determined from a verbal situation, which results in an equation, like in Example 4 where there was a flat fee of $150 to rent any number of bikes. In Example 1, the y-intercept is 150.
b. Y-intercept can be read from a table or graph. It is the point where x=0. In Example 4, it would be the point (0,150).
c. Y-intercept can be found by substituting the slope and one of the points of a line into the equation y=Mx+b and solving for b.
Finding the y-intercept is briefly explored in Moving Straight Ahead.
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