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4. Functions & Linear Relationships
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United States of AmericaTennessee
Algebra 1

4.07 Comparing linear and nonlinear functions

Lesson
Practice
Lesson

Concept summary

We can use key features to compare linear and nonlinear 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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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 nonlinear.

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 nonlinear.

Outcomes

A1.F.IF.A.1

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

A1.F.IF.A.2.A

Use function notation to evaluate functions for inputs in their domains, including functions of two variables.

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.*

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

A1.F.IF.C.9.A

Compare properties of two different functions. Functions may be of different types and/or represented in different ways.

A1.F.IF.C.9.B

Compare properties of the same function on two different intervals or represented in two different ways.

A1.F.LE.A.1.A

Know that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

A1.F.LE.A.1.B

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

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