standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
This activity formally introduces students to the absolute value function as the function that takes an input value and gives its distance from the origin as the output. Students see two different ways to represent this function algebraically: using the absolute value notation and using the cases notation:
\(A(x)=|x|\)
\(A(x)=\begin{cases} x,& x \geq 0 \\ \text{-}x, & x < 0 \end{cases}\)
The equation in cases notation focuses on what has to happen algebraically to an input value to give its distance from the origin. Different rules apply to different intervals of the domain.
As students work, look for those who plot points for the graph of function
\(A\)
and those who sketch two lines. Ask students who sketch different types of graphs to share during discussion.
Launch
Arrange students in groups of 2. Give students a few minutes of quiet work time, than time to share their response with their partner. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their verbal and written responses to the last question. Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by asking “Can you give a few examples?”, “How do you know?”, or “Can you demonstrate your thinking using the graph?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students understand different ways to represent an absolute value function.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Student Facing
The function
\(A\)
gives the distance of
\(x\)
from 0 on the number line.
Complete the table and sketch a graph of function
\(A\)
.
|       \(x\)       | \(A(x)\) |
|-------------------|----------|
|         8         |          |
|                   |   5.6    |
|      \(\pi\)      |          |
|    \(\frac12\)    |          |
|                   |    1     |
|         0         |          |
| \(\text-\frac12\) |          |
|        -1         |          |
|       -5.6        |          |
|                   |    8     |
[IMG]
Andre and Elena are trying to write a rule for this function.
Andre writes:
\(A(x)=\begin{cases} x,& x \geq 0 \\ \text-x, & x < 0 \\ \end{cases}\)
Elena writes:
\(A(x) = |x|\)
Explain why both equations correctly represent the function
\(A\)
.
Student Response
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Anticipated Misconceptions
Students who interpret the “
\(\text-x\)
” in Andre's rule to mean "negative
\(x\)
" (rather than "the opposite of
\(x\)
") may be unsure how to use that information. Ask students to evaluate the function for specific values of
\(x\)
and to write down each step. For instance, when
\(x\)
is -2,
\(A(\text{-}2)\)
is
\(\text{-}(\text{-}2)\)
, which is 2.
Activity Synthesis
Select previously identified students to display their graph of function
\(A\)
. Ask students who plotted only the ordered pairs in the table whether the pairs that are not in the table, if plotted, would also fall on the same two lines. Emphasize that
\(A\)
can be shown with two lines.
Next, ask students to share their response to the last question. Students may find Elena's rule easier to explain because of their work with absolute value in recent activities and struggle to explain Andre's rule.
We can reason about Andre's equation a couple of ways:
By thinking about rules: The equation is that of a piecewise function because different rules are applied to different parts of the domain to give the "distance from 0" as the output:
When the input
\(x\)
is positive or 0, its distance from 0 is that same number,
\(x\)
.
When the input
\(x\)
is negative, its distance from 0 is the opposite of the number,
\(\text-x\)
.
By using the graph: The two halves of the graph are lines with different slopes.
If we cover up the left side of the vertical axis and see only positive values of
\(x\)
, we see a line with a slope of 1 and that represents
\(y=x\)
.
If we cover up the right side and see only negative values of
\(x\)
, we see a line with a slope of -1 and that represents
\(y=\text{-}x\)
Explain to students that:
Function
\(A\)
is the
absolute value function
. It gives the distance of an input value from a certain value, the origin, in this case.
The graph of function
\(A\)
is a V shape with the two lines of points converging at
\((0,0)\)
, which is the minimum of the graph. We call this point the
vertex
of the graph. It is the point where the graph changes direction.
The absolute value function is also a piecewise function because different rules are applied to different parts of the domain to get the output.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: absolute value function, vertex, cases notation.
Supports accessibility for: Conceptual processing; Language


aligns: no