Toronto is a city at the border of the United States and Canada, just north of Buffalo, New York. Here are twelve guesses of the average temperature of Toronto, in degrees Celsius, in February 2017.
5
2
-5
3
0
-1
1.5
4
-2.5
6
4
-0.5
The actual average temperature of Toronto in February 2017 is 0 degrees Celsius.
Use this information to sketch a scatter plot representing the guesses, , and the corresponding absolute guessing errors, .
What rule can you write to find the output given the input?
14.2
Activity
The Distance Function
The function gives the distance of from 0 on the number line.
Complete the table with at least one possible value in each blank position, and sketch a graph of function .
8
5.6
1
0
-1
-5.6
8
Andre and Elena are trying to write a rule for this function.
Andre writes:
Elena writes:
Explain why both equations correctly represent the function .
14.3
Activity
Moving Graphs Around
Here are equations and graphs that represent five absolute value functions.
Notice that the number 2 appears in the equations for functions , and . Describe how the addition or subtraction of 2 affects the graph of each function.
Then, think about a possible explanation for the position of the graph. How can you show that it really belongs where it is on the coordinate plane?
14.4
Activity
More Moving Graphs Around
Here are five equations and four graphs.
Equation 1:
Equation 2:
Equation 3:
Equation 4:
Equation 5:
A
B
C
D
Match each equation with a graph that represents it. One equation has no match.
For the equation without a match, sketch a graph on the blank coordinate plane.
Use graphing technology to check your matches and your graph. Revise your matches and graphs as needed.
Student Lesson Summary
In a guessing game, each guess can be seen as an input of a function and each absolute guessing error as an output. Because absolute guessing error tells us how far a guess is from a target number, the output is distance.
Suppose the target number is 0.
We can find the distance of a guess, , from 0 by calculating . Because distance cannot be negative, what we want to find is , or simply .
If function gives the distance of from 0, we can define it with this equation:
Function is the absolute value function. It gives the distance of from 0 by finding the absolute value of .
The graph of function is a V shape with the two lines converging at .
We call this point the vertex of the graph. It is the point where a graph changes direction, from going down to going up, or the other way around.
We can also think of a function like as a piecewise function because different rules apply when is less than 0 and when is greater than 0.
Suppose we want to find the distance between and 4.
We can find the difference between and 4 by calculating . Distance cannot be negative, so what we want is the absolute value of that difference: .
If function gives the distance of from 4, we can define it with this equation:
Now suppose we want to find the distance between and -4.
We can find the difference of and -4 by calculating , which is equal to . Distance cannot be negative, so let's find the absolute value: .
If function gives the distance of from -4, we can define it with this equation:
Notice that the graphs of and are like that of , but they have shifted horizontally.
Glossary
absolute value
The absolute value of a number is its distance from 0 on the number line.
vertex (of a graph)
The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing, or vice versa. It is the highest or lowest point on the graph.
