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4. Modeling with Functions
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United States of America
Grades 9-12

4.01 Key features of functions

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Practice

Introduction

In Math 1, we learned about the key features of functions in lessons  3.02 Domain and range  and  3.04 Characteristics of functions  . We learned about other key features specific to exponential and quadratic functions in lessons  5.01 Exponential functions  and  10.01 Characteristics of quadratic functions  . Then in lesson  10.05 Comparing functions  , we used these features to compare various function types. These features will be important in our study of Math 2 concepts, so we will review them in this lesson.

Key features of functions

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 coordinate plane.

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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 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:

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 equals negative x plus 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 equals negative x plus 3 drawn in a coordinate plane. The point (0,3) is labeled y-intercept.
Vertex

The maximum or minimum point of a quadratic function or an absolute value function, written as an ordered pair.

The graph of a quadratic function that opens upward with a point plotted at its minimum labeled vertex.
A curve plotted in a four quadrant coordinate plane. The curve resembles the letter m with the left peak higher than the right peak. The left peak is labeled absolute maximum, right peak labeled relative maximum, and the trough labeled relative minimum.

  • Absolute maximum: The point with the largest y-value across the domain

  • Absolute minimum: The point with the smallest y-value across the domain

  • Relative maximum: The point with the largest y-value in a region of the domain

  • Relative minimum: The point with the smallest y-value in a region of the domain

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.

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Sections of functions can also display certain properties.

A connected region in the domain over which the output values become higher as the input values become higher is known as an increasing interval. Similarly, a connected region in the domain over which the output values (y-values) of a function become lower as the input values (x-values) become higher is known as a decreasing interval.

A graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The first and third segment are colored green and labeled with increasing interval. The second segment is colored blue. On the x axis, the intervals negative 5 to negative 2 and 0 to 4 are colored green. The interval negative 2 to 0 is colored blue.
Increasing intervals: (-5, -2)\cup (0,4)
A graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The first and third segment are colored green. The second segment is colored blue and labeled decreasing interval. On the x axis, the intervals negative 5 to negative 2 and 0 to 4 are colored green. The interval negative 2 to 0 is colored blue.
Decreasing interval: (-2,0)

Note that we did not use square brackets to include the endpoints of the intervals. This is because the function is not increasing or decreasing at the points of change between increasing and decreasing intervals. At these points, the function is considered to have a rate of change of zero.

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 graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The part of graph on the interval negative 4 to negative 1 on the x axis is colored green and labeled positive interval. The rest of the graph is colored blue.
Positive interval: (-4, -1)
A graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The part of graph on the interval negative 4 to negative 1 on the x axis is colored green. The rest of the graph is colored blue and labeled negative intervals.
Negative intervals: (-5,-4)\cup (-1, 4)

Examples

Example 1

The function shown in the table below represents a continuous function.

x-3.5-3-2.5-2-1.5-1-0.500.511.522.5
f\left(x\right)-4\frac{3}{8}36\frac{7}{8}87\frac{1}{8}52\frac{3}{8}0-1\frac{3}{8}-11\frac{7}{8}818\frac{1}{8}

Use the table to determine how many intercepts the function has over the interval -3.5<x<2.5.

Worked Solution
Create a strategy

The x-intercepts are the points where y=0, and the y-intercepts are the points where x=0. However, only one such point is listed in the table.

To determine if there are other intercepts, we need to look for places where the outputs switch from positive to negative or vice versa. Because the function is continuous, we know the graph must have crossed the x-axis somewhere between those two points.

Apply the idea

The point \left(0,0\right) is the only y-intercept, and it is also an x-intercept.

When x=-3.5, f\left(x\right) is negative. But when x=-3, f\left(x\right) is positive. Therefore, there must be an x-intercept between these values.

When x=1, f\left(x\right) is negative. But when x=1.5, f\left(x\right) is positive. Therefore, there must be an x-intercept between these values.

This means f\left(x\right) has 3 x-intercepts and 1 y-intercept.

Reflect and check
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We can plot the points in the table and connect them to visualize the graph of the function.

We can see that the graph confirms there is a zero between x=-3.5 and x=-3 and another zero between x=1 and x=1.5.

Including the origin, there are 3 x-intercepts and 1 y-intercept.

Example 2

Consider the function shown in the graph:

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a

State the coordinates of the vertex.

Worked Solution
Create a strategy

For this function, the vertex is the absolute minimum. Note the values on the x-axis change by 1, and the values on the y-axis change by 2.

Apply the idea

It appears the vertex is halfway between x=-1 and x=0 and halfway between y=-8 and y=-10. Therefore, the vertex is at \left(-\dfrac{1}{2},-9\right).

Reflect and check

Recall the axis of symmetry is a key feature of quadratic functions. The axis of symmetry is the line that passes through the middle of the parabola and the x-value of the vertex. So, the axis of symmetry for this parabola is x=-\dfrac{1}{2}.

b

State the positive and negative intervals.

Worked Solution
Create a strategy

The positive intervals are the domain values for which the parabola lies above the x-axis. We can see that this happens on the left side of the leftmost x-intercept and on the right side of the rightmost x-intercept. The function is negative between the intercepts.

Apply the idea

The function is positive on the intervals \left(-\infty, -3.5\right)\cup\left(2.5,\infty\right).

The function is negative on the interval \left(-3.5,2.5\right).

Reflect and check

Similar to increasing and decreasing intervals, the intercepts are not included in the interval since 0 is neither positive nor negative. That is why we used parentheses for the end values of the intervals.

c

State the domain and range in interval notation.

Worked Solution
Create a strategy

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.

Apply the idea

If we were to continue extending both ends of the function indefinitely, it would stretch upwards towards positive infinity on both sides. In addition, the ends would continue indefinitely toward the left and toward the right.

In part (b) we stated that the vertex is the point \left(-\dfrac{1}{2},-9\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 -9". In interval notation, this is:

  • Domain: \left(-\infty, \infty\right)

  • Range: \left[-9, \infty\right)

Reflect and check

In set-builder notation, we would write the domain and range as follows:

  • Domain: \left\{x\vert x\in\Reals\right\} or \left\{x\vert -\infty < x < \infty \right\}
  • Range: \left\{y\vert y\geq -9\right\}

Example 3

Consider the function shown in the graph:

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a

State the equation of the asymptote.

Worked Solution
Create a strategy

Notice how the function approaches the x-axis without ever reaching it:

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Apply the idea

This means that the x-axis is an asymptote for the function. The equation of the horizontal aymptote is y = 0.

Reflect and check

Asmptotes are lines, so they should always be stated as an equation. The equation for horizontal asymptotes is always of the form y=c, and the equation for vertical asymptotes is always of the form x=c where c is any real number.

b

Identify the intercepts.

Worked Solution
Create a strategy

In part (a), we found there is a horizontal asymptote on the x-axis. This means there will be no x-intercepts, so we only need to identify the y-intercept.

Apply the idea

The y-intercept of the function is at \left(0,1\right).

Reflect and check

Recall from Math 1 that all the exponential functions of the form f\left(x\right)=b^x will have a y-intercept at \left(0,1\right). This is because f\left(0\right)=b^0=1. Later in this unit, we will begin transforming functions, so we will see exponential functions that do not have a y-intercept at \left(0,1\right).

c

Describe the end behavior of the function.

Worked Solution
Create a strategy
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The end behavior refers to the left and right "ends" of the graph. We want to know what happens to the output values as the input values get increasingly smaller (to the left) and what happens to the output values when the input values get increasingly larger (to the right).

Apply the idea

As the input values decrease indefinitely, the function output values continue to increase indefinitely. So as x \to -\infty, f\left(x\right)\to \infty.

As the input values increase indefinitely, the function output values approach the asymptote at y=0. So as x \to \infty, f\left(x\right)\to 0.

Example 4

Consider the function graphed below.

A curve plotted in a four quadrant coordinate plane. The curve has turning points at (negative 3.5, 6), (negative 1, negative 3),(1.5, 0.5), (4, negative 4), and (6,0). Speak to your teacher for more details.
a

Determine the increasing and decreasing intervals.

Worked Solution
Create a strategy

To determine the increasing and decreasing intervals, we need to first find the x-values of the turning points. These values will be the endpoints of the increasing and decreasing intervals.

The turning points occur at x=-3.5,-1,1.5,4,6.

Apply the idea

Tracing the graph from the smallest x-values toward the largest x-values, we can see that the graph switches between increasing and decreasing at each of the turning points.

Increasing intervals: \left(-\infty, -3.5\right)\cup \left(-1,1.5\right)\cup \left(4,6\right)

Decreasing intervals: \left(-3.5, -1\right)\cup \left(1.5,4\right)\cup \left(6,\infty\right)

Reflect and check

Notice that we are not intersted in the y-values when listing the increasing and decreasing intervals. If we used the y-values, some of them would be listed in multiple intervals which would make the notation confusing. By only using the domain values, the notation is clear and the x-values do not appear in multiple intervals.

b

State any absolute and relative maxima and minima.

Worked Solution
Create a strategy

We have already identified the x-values of the turning points in part (a). Now, we need to identify the corresponding y-values and classify each as an absolute maximum, an absolute minimum, a relative maximum, or a relative minimum.

Apply the idea

The point with the largest y-value is \left(-3.5, 6\right). This is the absolute maximum.

The relative maxima are \left(1.5, 0.5\right) and \left(6,0\right).

The relative minima are \left(-1, -3\right) and \left(4, -4\right).

There is no absolute minimum since the end behavior of the function tends toward negative infinity.

Idea summary

The key features of functions include:

  • domain and range

  • x- and y-intercepts

  • maximum or minimum value(s)

  • vertex

  • axis of symmetry

  • end behavior

  • positive and negative intervals

  • increasing and decreasing intervals

  • asymptote(s)

Outcomes

F.IF.B.4

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

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