Tools for Instruction
Linear and Nonlinear Functions
Objective Determine if a function is linear or nonlinear by
examining if it fits the format y 5 mx 1 b.
Students have had many experiences with both linear and nonlinear functions. They have graphed functions,
evaluated them at various values, and used them to model real-world problems. They know that when graphed,
some equations form lines and some do not. Now they need to develop fluency in recognizing an equation
as linear or nonlinear. This will help them to recognize graphs of functions, decontextualize problems, and
understand the essential characteristics of functions.
In this activity, students will determine if a function is linear or nonlinear by determining if the equation is in the
form y 5 m x 1 b. The goal is for students to discern patterns in the equations and recognize the objects that the
equation is constructed from in order to determine its structure.
Step by Step 20–30 minutes
1 Model examining a nonlinear function.
• Ask the student to state the definition of a linear equation (an equation whose graph is a straight line).
• Explain that we want to determine if the equation y 5 4​ x2​ ​2 5 is linear, and we know that all linear equations
can be written in the form y 5 mx 1 b. We first need to make sure the equation is simplified so we can check
its form. Ask the student if we can combine any like terms or simplify the equation.
• Ask if the equation is in the form y 5 mx 1 b. Make sure the student sees that this equation is not linear
because the x variable has an exponent of 2, so the equation cannot be written in the form y 5 mx 1 b.
• Verify that the equation is not linear by graphing. Make a table showing x and y values for x 5 22, 21, 0, 1, 2,
and then graph the points.
• Ask if the graph shows a straight line. Remind the student that on a straight line, each time x increases by 1,
y changes by the same amount. Ask the student if that pattern holds in this graph. (no)
2 Model examining a linear equation.
• Remind the student again that we are checking to see if the equation fits the pattern y 5 mx 1 b. If it does,
we know the equation is linear.
• Ask if the equation can be simplified. (Yes, it can be simplified to y 5 5 x 2 10)
• Ask: Is y 5 5 x 2 10 in the form y 5 mx 1 b? What is m? (5) What is b? (210) We have minus 10 in our equation,
and linear equations have plus b. Is this OK? Why? (This is OK because b itself can be a negative number; we can
write this equation as y 5 5 x 1 ​( 210 )​, where b is equal to 210.)
• Verify that the equation is linear by graphing. Make a table showing x and y values for x 5 22, 0, 2, 4, and
then graph the points.
• Ask the student if y changes at a constant rate each time x increases by 1. Remind the student that this is
another way to identify linear functions.
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Algebra and Algebraic Thinking
I Level 8 I Linear and Nonlinear Functions I Page 1 of 2
i-Ready Tools for Instruction
3 Repeat with other equations.
• Depending on the student, make up appropriate equations or select from y 5 28 x, y 5 x​( x 2 2 )​, y 5 7 1 p x,
y 5 5 x 1 2 1 3 x 2 1, or y 5 ​x3​ ​1 x 1 3.
• For each equation, ask the student if the equation can be simplified and if it is in the form y 5 mx 1 b.
• If the equation is linear, ask the student to identify m and b and describe how those values are shown in the
graph of the function.
Check for Understanding
Ask the student to write two equations that are linear. For each equation, have the student identify m and b and
explain how the graph shows these values. Then have the student write two equations that are not linear and
have him explain why the equations are not linear.
For the student who struggles, use the chart below to help pinpoint where extra help may be needed.
If you observe…
the student may…
Then try…
the student sees x2 and 2 x as
equivalent expressions
not understand exponents.
reviewing how to read exponents
and stressing the relationship
between how they are read and
what they mean. Then reinforce
the concept by having the student
explain to you the meaning of x4
and 4 x.
the student has difficulty
simplifying expressions
not understand the Distributive
Property.
using simple manipulatives or
diagrams to emphasize that
3 ​( x 1 1 )​means 3 groups of
x 1 1 which is the same as
3 x 1 3. Then have the student
use the Distributive Property to
rewrite 5 ​( x 1 3 )​.
the student does not recognize
equations in the form y 5 mx as
linear
not understand that b can be 0 in
a linear equation.
pointing out that we could write
y 5 4x as y 5 4x 1 0, but that we
don’t because adding 0 doesn’t
change the value.
the student has difficulty coming
up with a nonlinear equation
not realize that linear equations
never contain a variable raised to
a power of 2 or greater.
referring the student back to the
examples of nonlinear equations.
Ask the student to identify how
these equations are different from
linear equations.
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©2012 Curriculum Associates, LLC
Algebra and Algebraic Thinking
I Level 8 I Linear and Nonlinear Functions I Page 2 of 2