Topics
1. Analyzing Functions
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United States of AmericaTennessee
Algebra 2

1.01 Key features and functions

Lesson
Practice
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Concept summary

A relation in mathematics is a set of pairings between input and output values. A relation in which each input corresponds to exactly one output is known as a function. Input-output pairs of a function or relation are often written as coordinates in the form \left(x, y\right), especially when relating to a graphical representation, which can then be graphed on the axes (of a graph).

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The set of all possible inputs is called the domain, while the set of all possible outputs is called the range.

An interval is a set of all numbers which lie between two values. The domains and ranges of functions can sometimes be represented as intervals (or combinations of intervals) using interval notation or set-builder notation.

There are many types of functions. Some of the most common include:

Linear function

A function that has a constant rate of change. A linear function can be written in the form y = mx + b

Quadratic function

A polynomial function of degree 2. A quadratic function can be written in the form y = ax^2 + bx + c where a \neq 0

Exponential function

A function that has a constant percent rate of change. An exponential function can be written in the form y = ab^x where a \neq 0 and b>0

Functions are frequently represented graphically on a coordinate plane. Graphs of quadratic functions are given the special name parabola.

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Linear function: line
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Quadratic function: parabola
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Exponential function
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Neither linear, quadratic, nor exponential

There are many key features that can be used to identify, describe, and analyze functions. These key features include the domain and range, as well as the following:

Asymptote

A line that a curve approaches as one or both of the variables in the equation of the curve approach infinity.

A decreasing exponential function approaching but never touching a dashed horizontal line labeled asymptote
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.

Vertex

The point on a function that obtains the maximum or minimum function output value, written as an ordered pair. Not every function type has a vertex.

The graph of a quadratic function that opens upward with a point plotted at its minimum labeled vertex
x-intercept

A point where a line or graph crosses the x-axis. The value of y is 0 at this point. A function can have any number of x-intercepts.

The line y=-x+3 drawn in a coordinate plane. The point (3,0) is labeled x-intercept
y-intercept

A point where a line or graph crosses the y-axis. The value of x is 0 at this point. A function can have at most one y-intercept.

The line y=-x+3 graphed in the coordinate plane. The point (0,3) is labeled y-intercept

Sections of functions can also display certain properties.

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If the output values (y-values) of a function become lower as the input values (x-values) become higher, the function is said to be a decreasing function.

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If the output values become higher as the input values become higher, the function is said to be an increasing function.

A connected region in the domain over which the function values all lie above the x-axis is known as a positive interval. Similarly, a connected region in the domain over which the function values all lie below the x-axis is known as a negative interval.

A broken line graph highlighting the part of the graph above the x-axis as the positive interval
Positive interval
A broken line graph highlighting the parts of the graph below the x-axis as the negative interval
Negative interval

Worked examples

Example 1

Consider the function shown in the table below:

x012345
f\left(x\right)-7-315913
a

Determine whether the values represent a linear, quadratic, or exponential function:

Solution

A linear function has a constant rate of change, a quadratic function has a linear rate of change, and an exponential function has a constant percent rate of change. To check the rate of change, we calculate the differences between the function values:

\displaystyle f\left(1\right) - f\left(0\right)\displaystyle =\displaystyle -3 - \left(-7\right) = 4
\displaystyle f\left(2\right) - f\left(1\right)\displaystyle =\displaystyle 1 - \left(-3\right) = 4
\displaystyle f\left(3\right) - f\left(2\right)\displaystyle =\displaystyle 5 - 1 = 4
\displaystyle f\left(4\right) - f\left(3\right)\displaystyle =\displaystyle 9 - 5 = 4
\displaystyle f\left(5\right) - f\left(4\right)\displaystyle =\displaystyle 13 - 9 = 4

So we can see that the rate of change is constant, and therefore the function is linear.

b

Identify the y-intercept of the function.

Solution

The y-intercept of a function is the point on the function where x = 0.

Looking at the table, we can see that x = 0 corresponds to f\left(x\right) = -7. So the y-intercept is the point \left(0, -7\right).

Example 2

Consider the function shown in the graph:

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a

State the coordinates of the y-intercept

Solution

The y-intercept of the function is the point at which it crosses the y-axis:

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We can see that this point is at \left(0, 1\right).

b

State the coordinates of the vertex

Solution

The vertex of the function is the point at which the function reaches its maximum or minimum value. In this case, the vertex is a minimum point:

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We can see that this point is at \left(-2, -3\right).

c

State the domain and range of the function, using interval notation

Approach

Remember that the domain of the function is the set of all possible input values, which are the x-values that correspond to points on the graph.

Similarly, the range of the function is the set of all possible output values, which are the y-values that correspond to points on the graph.

Solution

If we were to continue extending both ends of the function indefinitely, it would stretch upwards towards positive infinity on both sides.

In part (b) we stated that the vertex is the point \left(-2, -3\right) and we can see that it is the minimum point on the function.

So the domain is "all real values" and the range is "all values greater than or equal to -3". In interval notation, this is:

  • Domain: \left(-\infty, \infty\right)
  • Range: \left[-3, \infty\right)

Example 3

Consider the function shown in the graph:

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a

State the equation of the asymptote

Solution

The function approaches the x-axis without ever reaching it:

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This means that the x-axis, which has the equation y = 0, is an asymptote for the function.

b

State how many x-intercepts the function has

Solution

This function approaches the x-axis without ever reaching it, so it has no x-intercepts.

c

State the domain and range of the function, using interval notation

Approach

Remember that the domain of the function is the set of all possible input values, which are the x-values that correspond to points on the graph.

Similarly, the range of the function is the set of all possible output values, which are the y-values that correspond to points on the graph.

Solution

If we were to continue extending both ends of the function indefinitely, it would stretch upwards towards positive infinity on the left, while on the right it will get closer and closer to zero without touching the x-axis.

So the domain is "all real values" and the range is "all values greater than 0". In interval notation, this is:

  • Domain: \left(-\infty, \infty\right)
  • Range: \left(0, \infty\right)
d

Describe the end behavior of the function

Solution

To the right, as x \to \infty, the function output values approach 0 (as the function approaches its asymptote).

To the left, as x \to -\infty, the function output values continue to increase, i.e. f\left(x\right) \to \infty.

Outcomes

A2.N.Q.A.1

Use units as a way to understand real-world problems.*

A2.F.IF.A.1

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

A2.F.IF.A.2

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

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP4

Model with mathematics.

A2.MP5

Use appropriate tools strategically.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

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