Topics
4. Functions & Linear Relationships
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United States of AmericaFL
Grade 912

4.08 Comparing linear and nonlinear functions

Lesson
Practice
Lesson

Concept summary

We can use key features to compare linear and non-linear functions. Some additional key features that we might look at are:

End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that each end obtains or approaches

Positive interval

A connected region of the domain in which all function values lie above the x-axis

Negative interval

A connected region of the domain in which all function values lie below the x-axis

Worked examples

Example 1

Consider the two functions shown in the graphs below.

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y
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a

State the intercepts of each function.

Solution

Both functions have a y-intercept at \left(0, -1\right).

Also, both functions have an x-intercept at \left(1, 0\right).

The second function has an additional x-intercept at \left(-1,0 \right).

b

Compare the end behavior of the two functions.

Solution

On the right side, both functions take larger and larger positive values as x gets further from zero. That is, as x \to \infty, y \to \infty for both functions.

On the left side, the first function takes larger and larger negative values as x gets further from zero. That is, as x \to -\infty, y \to -\infty for the first function.

On the other hand, the second function takes larger and larger positive values as x gets further from zero on the left side. That is, as x \to -\infty, y \to \infty for the second function.

Reflection

Note that although both functions tend towards infinity to the right, the way they do so is different. The first function increases at a constant rate, while the second function increases at an increasing rate.

c

State the interval(s) over which each function is positive or negative.

Solution

The first function is positive for x > 1 and negative for x < 1.

The second function is positive for both x > 1 and x < -1, and is negative for -1 < x < 1.

d

Determine whether each function is linear or non-linear.

Approach

A linear function has a constant rate of change. It also has no turning points, with at most one x-intercept.

Solution

The first function is linear, while the second function is non-linear.

Outcomes

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

MA.912.AR.5.6

Given a table, equation or written description of an exponential function, graph that function and determine its key features.

MA.912.F.1.5

Compare key features of linear functions each represented algebraically, graphically, in tables or written descriptions.

MA.912.F.1.6

Compare key features of linear and nonlinear functions each represented algebraically, graphically, in tables or written descriptions.

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