Topics
1A. Polynomial and Rational Functions
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AP Precalculus

1.1 Change in tandem

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Lesson

Introduction

Learning objectives

  • 1.A.1 Describe how the input and output values of a function vary together by comparing function values.
  • 1.B.1 Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.

Inputs and outputs of functions

A function is a special mathematical relationship that pairs each input value from a set, called the domain, with exactly one output value from another set, called the range. Think of a function as a machine that takes an input, processes it according to a specific rule, and produces an output. We usually use the term independent variable for the input value and dependent variable for the output value, as the output depends on the input.

function

A relation in which each input corresponds to exactly one output

domain

The set of all possible input values (x-values) for a relation

range

The set of all possible output values (y-values) for a relation

A function is a rule that tells us how to calculate the output value for a given input value. A function rule tells us how the input and output values vary together, or covary. We can represent function rules in different ways, such as graphs, tables, formulas, or verbal descriptions.

In a graph, we can see how a function behaves over its domain.

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When a function is increasing over an interval, it means that as the input values increase, the output values also increase. We can represent this mathematically by saying that for all input values a and b in an interval, if a < b, then f\left(a\right)<f\left(b\right).

For example, in the graph shown f\left(-2\right)=3 and f\left(3\right)= 6. Since 3<6 the function is increasing over the interval \left(-2,3\right).

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On the other hand, when a function is decreasing over an interval, it means that as the input values increase, the output values decrease. We can represent this mathematically by saying that for all input values a and b in an interval, if a < b, then f\left(a\right)>f\left(b\right).

For example, in the graph shown f\left(-2\right)=6 and f\left(3\right)= 3. Since 6>3 the function is decreasing over the interval \left(-2,3\right).

Examples

Example 1

Given the function f\left(x\right) = x^2 + 3x - 4, create a table of values for x = -2, -1, 0, 1, \text{ and } 2, and analyze how the output values change as the input values increase.

Worked Solution
Create a strategy

To create a table of values, we will substitute each of the given input values into the function and calculate the corresponding output values. Then, we will analyze the table to determine how the output values change as the input values increase.

Apply the idea

For each value we will substitute into the function f\left(x\right) = x^2 + 3x - 4:

For x=-2: we get f\left(-2\right) = \left(-2\right)^2 + 3\left(-2\right) - 4 = 4 - 6 - 4 = -6

For x=-1: we get f\left(-1\right) = \left(-1\right)^2 + 3\left(-1\right) - 4 = 1 - 3 - 4 = -6

For x=0: we get f\left(0\right) = \left(0\right)^2 + 3\left(0\right) - 4 = 0 + 0 - 4 = -4

For x=1: we get f\left(1\right) = \left(1\right)^2 + 3\left(1\right) - 4 = 1 + 3 - 4 = 0

For x=2: we get f\left(2\right) = \left(2\right)^2 + 3\left(2\right) - 4 = 4 + 6 - 4 = 6

From the table of values, we can observe that as x increases from -2 to -1, the output value remains the same at -6.

As x increases from -1 to 0, the output value increases from -6 to -4.

As x increases from 0 to 1, the output value increases from -4 to 0.

Finally, as x increases from 1 to 2, the output value increases from 0 to 6.

Overall, for the part of the function reflected in the table, as the input values increase, the output values also increase, except for the interval from x=-2 to x=-1, where the output value remains constant.

Reflect and check

Though it appears that the function outputs remain constant over the interval x=-2 to x=-1, by graphing the function we can see what is really happening.

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The minimum of the function is located between x=-2 to x=-1.

Example 2

Jack is walking along a curved path. His distance from the starting point is represented by the function d\left(t\right) = 3t^2 - 2t, where d\left(t\right) is the distance in meters and t is the time in hours.

a

Identify the independent and dependent variables.

Worked Solution
Create a strategy

Recall that an independent variable is the variable that can be controlled or manipulated, while the dependent variable is the variable that depends on the independent variable.

Apply the idea

In this example, the distance Jack travels, d\left(t\right), depends on the amount of time, t, he has spent walking.

So the dependent variable is Jack's distance, d\left(t\right), and the independent variable is the time, t.

Reflect and check

The relationship between independent and dependent variables is critical in understanding the behavior of functions. It's essential to recognize that the dependent variable relies on the independent variable for its value. In this problem, the distance that Jack travels depends on the time elapsed, illustrating the cause-and-effect relationship between the two variables.

b

Determine if Jack's distance from the starting point is increasing or decreasing between t = 2 hours and t = 3 hours and explain your reasoning.

Worked Solution
Create a strategy

To determine whether the function is increasing or decreasing between two specific input values, we can compare the output values at these input values. If the output value is greater at the later time, the function is increasing. If the output value is smaller at the later time, the function is decreasing.

Apply the idea
  1. Calculate the distance at t=2 hours: d\left(2\right)=3\left(2\right)^2-2\left(2\right)=8 meters.
  2. Calculate the distance at t=3 hours: d\left(3\right)=3\left(3\right)^2-2\left(3\right)=21 meters.
  3. Compare the output values: Since d\left(3\right)>d\left(2\right), Jack's distance from the starting point is increasing between t = 2 hours and t = 3 hours.
Reflect and check

The function d\left(t\right) = 3t^2 - 2t represents a quadratic function. By graphing, we can confirm that there is an increasing interval between t = 2 and t = 3.

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Idea summary
  • Functions map input values to unique output values.
  • A function is increasing when output values increase as input values increase, and decreasing when output values decrease as input values increase.
  • To analyze functions, compare output values at different input values or observe a graph.

Graphs of functions

The graph of a function displays a set of input-output pairs and shows how the values of the function's input and output values vary. To create a graph based on a verbal description, begin by carefully analyzing the description to understand the relationship between the variables involved. Pay attention to how the variables change with respect to each other, whether they increase, decrease, or fluctuate in some pattern. Identify the independent variable (input) and dependent variable (output) from the description.

Once the relationship between the variables is clear, determine the appropriate scale for the axes based on the range of input and output values. Generate a set of input values within the relevant range and use the function rule or the verbal description to find their corresponding output values. Plot these input-output pairs on the graph to create a visual representation of the relationship between the two variables.

Hot Air Balloon Ride
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Consider the scenario of a hot air balloon ride. The altitude of the hot air balloon changes as time passes. The hot air balloon starts at ground level, quickly rises to a high altitude, maintains that altitude for a short time, and then gradually descends back to the ground. This relationship between altitude and time is shown on the graph.

Once a function is graphed, it is easy to identify features of the function. One feature that can be useful to identify is concavity. Concavity relates to the curve of the graph.

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A graph is concave up on intervals in which the rate of change is increasing. This means that as we move from left to right along the graph, the slope of the tangent line to the graph increases (becomes more positive).

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A graph is concave down on intervals in which the rate of change is decreasing. In this case, as we move from left to right along the graph, the slope of the tangent line to the graph decreases (becomes more negative).

The point of inflection of a function is the point where the concavity changes from concave up to concave down, or from concave down to concave up. The rate of change of this point is faster than the rate of change of the points near it.

Another feature that is extremely helpful to identify are the zeros of a function. Zeros can be found at the x-intercepts, or where the output of the function is zero (this is why they are called zeros).

Zeros are useful to know because they have many real world applications such as when an object lands on the ground or when a company breaks even on their investment. Zeros can be found by identifying the x-intercepts, from a graph or by setting the equation of the function equal to 0 and solving for the independent variable.

Examples

Example 3

The following graph represents the water consumption for a household. The water bill B\left(x\right) in dollars for a household is based on x hundred gallons of water consumed during a month.

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a

If a household consumes 500 gallons of water during a month, what will be their water bill?

Worked Solution
Create a strategy

If a household consumes 500 gallons then they have consumed 5 hundred gallon units of water. So in this situation we need to find x=5 on the x-axis to determine the bill.

Apply the idea

Find x=5 on the x-axis. Then locate the output value of the function on the B\left(x\right)-axis (or y-axis).

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The output is 70 which means the family's water bill for a month where they consume 500 gallons of water is \$70.

b

If a household receives a water bill of \$ 95, approximately how many hundred gallons of water did they consume during the month?

Worked Solution
Create a strategy

The water bill is represented by the vertical, B\left(x\right)-axis so we need to start by finding \$95 on that axis. Then we can identify the input that corresponds with an output of \$ 95.

Apply the idea

Find B\left(x\right)=95 on the B\left(x\right)-axis. Then locate the input value of the function on the x-axis.

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The input is approximately 7.5 which means when the family's water bill is \$ 95 the family consumed 750 gallons of water for the month.

c

For which values of x does the water bill increase the fastest? What does this mean in the context of this problem?

Worked Solution
Create a strategy

We can look at the rate of change, or slope, of each segment of the function to determine which segment is increasing the fastest.

Apply the idea

It appears the first segment is the steepest, but to confirm we can find the slope of each segment by using the ratio of the vertical change to the horizontal change (also called rise over run).

Left segment: \dfrac{\text{rise}}{\text{run}}=\dfrac{\$ 30}{2 \text{ hundred gal}}=\$ 15 per hundred gallons of water.

Middle segment: \dfrac{\text{rise}}{\text{run}}=\dfrac{\$ 40}{4 \text{ hundred gal}}=\$ 10 per hundred gallons of water.

Right segment: \dfrac{\text{rise}}{\text{run}}=\dfrac{\$ 10}{2 \text{ hundred gal}}=\$ 5 per hundred gallons of water.

We can see that the left segment is in fact the steepest. So the water bill increases the fastest for values of x between 0 and 4. This means that the cost of water increases most rapidly for families consuming less than 400 gallons of water per month.

Example 4

A small town is hosting a fair, and a hot air balloon ride is one of the attractions. The balloon's height above the ground over time can be described as follows: At t = 0, the balloon is on the ground. As time passes, the balloon slowly rises for the first minute and a half and then rises very quickly until it reaches its highest point at 50 feet after 2 minutes. It then descends steadily until it reaches the ground after 4 minutes.

a

Create a graph that could represent the height of the hot air ballon as a function of time. Choose appropriate axes labels and scale for the graph.

Worked Solution
Create a strategy

We can find input-output pairs for different times, t, and their corresponding heights, h\left(t\right), based on the verbal description. We have information about the height at times t=0, t=2, and t=4. Then we can estimate what the function looks like between these points based on the description of the rate of change.

Apply the idea

Start by graphing the points \left(t,h\left(t\right)\right) identified in the verbal description.

The points are: A\left(0,0\right), B\left(2,50\right), C\left(4,0\right)

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The label for the horizontal, t-axis is minutes (min) and the label for the vertical f\left(t\right)-axis is feet (ft). Many different scales are appropriate as long as the maximum and minimum values needed for each variable are shown.

Now using the descriptions of the rate of change between these points we can estimate what the graph might look like on those intervals.

Between points A and B the balloon is described as rising slowly for the first 1.5 minutes follow by a quick increase. So the graph should represent a gradual rate of change (that is not steep) until 1.5 minutes where it should get very steep.

Between points B and C the balloon is described as descending steadily. This section of the graph can be represented by a linear function so that the rate of change is constant over the interval from point B to C. The following graph is one example of what this scenario might look like.

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b

Identify the zeros of the function and explain what they represent in this context.

Worked Solution
Create a strategy

The zeros correspond to the times when the hot air balloon is on the ground. On the graph these points are represented by the t-intercepts.

Apply the idea

Looking at the graph from part (a) we can see that the zeros are t=0 and t=4 minutes. This represents the two times when the balloon is on the ground.

Example 5

Consider the following graph of f\left(x\right):

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Is f\left(x\right) increasing, decreasing, or constant at x=4. Explain.

Worked Solution
Create a strategy

We can look at the slope of the tangent line at x=4 to see if it is positive or negative.

Apply the idea

Visualize the tangent line to the graph at x=4.

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The slope of the tangent line is negative so the function is decreasing at x=4.

Reflect and check

We can also see on the graph that, for the interval containing x=4, as the input (x) values increase, the output (y) values decrease, which also tells us the function is decreasing at x=4.

b

Is the rate of change of f\left(x\right) increasing, decreasing, or constant at x=4. Explain.

Worked Solution
Create a strategy

Recall the concavity of a function tells us where the rate of change is increasing or decreasing. A function is concave up on intervals where the rate of change is increasing and concave down on intervals where the rate of change is decreasing.

Apply the idea

At x=4 the function is concave down. So the reate of change is decreasing.

c

State the interval where f\left(x\right) is concave up.

Worked Solution
Apply the idea

f\left(x\right) is concave up on the approximate interval - \infty \lt x \lt 2.5.

Reflect and check

For a cubic function, the concavity switches at the point of inflection.

d

State the interval where f\left(x\right) is concave down.

Worked Solution
Apply the idea

f\left(x\right) is concave down on the approximate interval 2.5 \lt x \lt \infty.

Idea summary
  • The graph of a function represents the relationship between input and output values.
  • Graphs intersect the x-axis at the zeros of the function.
  • A function is concave up when the rate of change is increasing.
  • A function is concave down when the rate of change is decreasing.

Outcomes

1.1.A

Describe how the input and output values of a function vary together by comparing function values.

1.1.B

Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.

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