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5. Linear Functions
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Algebra, Functions, and Data Analysis

5.01 Characteristics of linear functions

Lesson
Practice

Calculating slope

Recall the slope of a line is a value that describes the line's steepness.

Slope

The ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

We can find the slope of a line by identifying the vertical and horizontal change:\text{slope}=\dfrac{\text{change in }y}{\text{change in }x} = \dfrac{\text{rise}}{\text{run}}

There are four types of slope:

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Zero
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Undefined

Notice that lines with a positive slope rise from left to right, while lines with a negative slope fall from left to right. Lines with zero slope are horizontal, and lines with an undefined slope are vertical.

We can find the slope of a line from its graph by creating a slope triangle. Consider points A and B on the line:

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From point A to point B, we move 9 units down and 3 units to the right to create lines that form the sides of a triangle.

The "rise" or vertical change is -9, and the "run" or horizontal change is +3.

So we can write the slope as the fraction \dfrac{-9}{3}, which can be simplified to \dfrac{-3}{1} or -3.

Points A and B are not the only points we could have used to find the slope. We can use any two points on a line; however, choosing points with integer coordinates can make counting the slope much easier. Consider points C and D instead.

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With this new triangle, we move 3 units down and 1 unit to the right.

The "rise" or vertical change is -3, and the "run" or horizontal change is +1.

So we can write the slope as the fraction \dfrac{-3}{1} or -3. This gives the same slope, -3, but simplified immediately.

If we pick two points that are so close together that no other points with integer coordinates lie between them, we will get a slope that is already simplified.

Finding the ratio of the rise and run of the line works when it is easy to see the graph, with clearly marked points on the line. We can extend this thinking to use the coordinates of two points and construct a general formula.

A line from point (x subscript 1, y subscript 1) to point (x subscript 2, y subscript 2) is drawn on the coordinate plane. The vertical distance from the x-axis to point (x subscript 1, y subscript 1) is labeled y subscript 1. The vertical distance from the x-axis to point (x subscript 2, y subscript 2) is labeled y subscript 2.

The rise of the line is found with points that lie on a vertical line. These points share the same x-value. The distance between them is the difference in the y-values.\text{rise} = y_2 - y_1

A line from point (x subscript 1, y subscript 1) to point (x subscript 2,y subscript 2) is drawn on the coordinate plane. The horizontal distance from the y-axis to point (x subscript 1, y subscript 1) is labeled x subscript 1. The horizontal distance from the y-axis to point (x subscript 2,y subscript 2) is labeled x subscript 2.

The run of the line is found with two points that lie on a horizontal line. These points share the same y-value. The distance between them is the difference in the x-values.\text{run} = x_2 - x_1

We find the slope by using the formula:

\displaystyle m= \dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}
\bm{m}
slope
\bm{\left(x_1,y_1\right)}
a point on the line
\bm{\left(x_2,y_2\right)}
a second point on the line

Examples

Example 1

Consider the points A, B, and C.

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a

Complete the directions to move from point A to point B:

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

Find the difference in the y-values (rise) between points A and B, and the difference between their x-values (run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 8-0Subtract the y-coordinates of points A and B
\displaystyle =\displaystyle 8\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 4-0Subtract the x-coordinates of points A and B
\displaystyle =\displaystyle 4\text{ units}Evaluate

From A, move 8 units up, and 4 units to the right.

b

Express the movement from part (a) as a simplified ratio of vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula, m=\dfrac{\text{rise}}{\text{run}}, then convert the fraction to a ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac84Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac{2}{1}Simplify
\displaystyle =\displaystyle 2\text{:}1Express as a ratio
c

Complete the directions to move from point A to point C.

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

Find the difference in the y-values (rise) between points A and C, and the difference between their x-values (run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 12-0Subtract the y-coordinates of points A and C
\displaystyle =\displaystyle 12\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 6-0Subtract the x-coordinates of points A and C
\displaystyle =\displaystyle 6\text{ units}Evaluate

From A, move 12 units up, and 6 units to the right.

d

Express the movement from the previous part as a simplified ratio of vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula, m=\dfrac{\text{rise}}{\text{run}}, then convert the fraction to a ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac{12}{6}Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac{2}{1}Simplify
\displaystyle =\displaystyle 2\text{:}1Express as a ratio

Example 2

Find the slope of the line in the graph.

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Worked Solution
Create a strategy

To find the slope of the line, use the formula: \text{Slope}=\dfrac{\text{rise}}{\text{run}}.

Apply the idea

From point A(-2,\,10) to point B(2,\,2), we move 8 units down and 4 units to the right.

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The ratio of the rise to the run is \dfrac{-8}{4}, which simplifies to -2.

The slope of the line is m=-2.

Example 3

The table shows the cost for different amounts of gasoline.

\text{Number of gallons }(x)010203040
\text{Cost of gasoline }(y)026.4052.8079.20105.60

How much does gasoline cost per gallon?

Worked Solution
Create a strategy

The cost per gallon of gasoline is the unit rate. This is the ratio of the change in cost (y) to the change in the number of gallons (x).

We can find this unit rate using the slope formula, m=\dfrac{y_2-y_1}{x_2-x_1}, with any two input-output pairs in the table.

Apply the idea
\displaystyle \text{Cost per gallon}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the slope formula
\displaystyle =\displaystyle \dfrac{26.40-0}{10-0}Substitute two input-output pairs
\displaystyle =\displaystyle \dfrac{26.40}{10}Evaluate the subtraction
\displaystyle =\displaystyle \$2.64Evaluate the division
Reflect and check

The relationship between the cost of gasoline and the number of gallons purchased can be modeled by the equation y=2.64x. Graphing this line shows that all the points in the table lie on the line.

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Example 4

Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.

The graph shows the amount of medication in the bloodstream for the liquid form.

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The table shows the results for the capsule form.

\text{Time (min)}, t\text{Amount in} \\ \text{blood (mg)}, A
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a

At what rate, in milligrams per minute, is the liquid form absorbed?

Worked Solution
Create a strategy

Choose any two points that lie on the line and use the formula for slope.

Apply the idea

Using the points (0,\,0)and (2,\,8) that lie on the line:

\displaystyle \text{Rate of liquid absorption}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the formula for slope
\displaystyle =\displaystyle \dfrac{8-0}{2-0}Substitute the values
\displaystyle =\displaystyle 4\text{ mg/min}Evaluate
b

At what rate, in milligrams per minute, is the capsule form absorbed?

Worked Solution
Create a strategy

Choose any two points from the table and use the slope formula.

Apply the idea

Using the points (4,\,24.6) and (7,\,42.3) from the table:

\displaystyle \text{Rate of capsule absorption}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the formula for slope
\displaystyle =\displaystyle \dfrac{42.3-24.6}{7-4}Substitute the values
\displaystyle =\displaystyle 5.9\text{ mg/min}Evaluate
c

In which form is the medication absorbed more rapidly?

Worked Solution
Create a strategy

The medication that is absorbed more rapidly will have a higher rate.

Apply the idea

5.9\text{ mg/min} \gt 4\text{ mg/min}Comparing the two rates, the capsule form is absorbed more rapidly than the liquid form as it has a higher rate of absorption of 5.9\text{ mg/min}.

d

Which form of medication has a higher amount in the blood at t=5 minutes?

Worked Solution
Create a strategy

First, find the amount for each form at t=5. For the liquid form, we can find the amount from the graph or its equation. For the capsule form, we can use the rate found previously to find its equation.

The equation for the liquid form is A = 4t.

The rate for the capsule is 5.9 \text{ mg/min}. Using point (4, 24.6), its equation is A - 24.6 = 5.9(t - 4), which simplifies to A = 5.9t + 1.

Apply the idea

For the liquid form at t=5, the amount is A = 4(5) = 20 \text{ mg}.

For the capsule form at t=5, the amount is A = 5.9(5) + 1 = 30.5 \text{ mg}.

Comparing the amounts: 30.5 \text{ mg} > 20 \text{ mg}.

The capsule form has a higher amount in the blood at t=5 minutes.

Idea summary

The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).\text{slope}=\dfrac{\text{change in }y}{\text{change in }x} = \dfrac{\text{rise}}{\text{run}}

The slope of a line is found using the formula:

\displaystyle m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}
\bm{(x_{1},\,y_{1})}
the coordinates of one point on the line
\bm{(x_{2},\,y_{2})}
the coordinates of a second point on the line

We can find the slope in different ways depending on how it is represented:

  • From a graph: Count the slope as the rise over the run, or find two points and use the slope formula.
  • From a table: Find the difference in two consecutive outputs and divide by the difference in the corresponding inputs, or find two input-output pairs and use the slope formula.
  • From a description: The slope is the rate given in the problem.

Key features of linear function graphs

Exploration

Look at these graphs. Think about where they cross the axes and what values are possible for x and y.

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Graph B (Height of a candle over time)
  1. Look at Graph A. At what point does the line cross the vertical axis (y-axis)? What is the x-coordinate of this point?
  2. At what point does Graph A cross the horizontal axis (x-axis)? What is the y-coordinate of this point?
  3. Does Graph A extend infinitely in both horizontal directions? What does this suggest about the possible x-values (domain)?
  4. Does Graph A extend infinitely in both vertical directions? What does this suggest about the possible y-values (range)?
  5. Now look at Graph B, which represents the height of a burning candle over time. Does this graph extend infinitely in all directions? Why or why not? What might be a reasonable domain and range for this situation based on the graph?

Besides slope, several other key features help us understand the graph of a linear function.

y-intercept

The point where the graph crosses the y-axis. The x-coordinate of this point is always 0. It often represents an initial value or starting point in context.

x-intercept

The point where the graph crosses the x-axis. The y-coordinate of this point is always 0.

Zero

The x-value where the function's output (y-value) is 0. This is the x-coordinate of the x-intercept.

Domain

The set of all possible input values (x-values) for the function.

Range

The set of all possible output values (y-values) for the function.

We can identify the intercepts directly from the graph by looking where the line crosses the axes.

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For the function graphed, the y-intercept is (0, -3), as the line crosses the y-axis at y=-3.

The x-intercept is (2, 0), as the line crosses the x-axis at x=2.

The zero of the function is x=2, because that is the x-value when y=0.

For most linear functions graphed on a full coordinate plane, the line extends infinitely in both directions. This means the domain (all possible x-values) includes all real numbers, written in interval notation as (-\infty, \infty).

Similarly, unless it is a horizontal line, the range (all possible y-values) also includes all real numbers, (-\infty, \infty). For a horizontal line like y=3, the domain is (-\infty, \infty) but the range is just the single value \{3\}.

However, in real-world contexts, the domain and range are often restricted. For example, if x represents time, the domain might be [0, \infty) because time cannot be negative. If y represents the number of items, the range might be restricted to non-negative integers or just non-negative values [0, \infty). A graph representing a specific event might be a line segment, with both domain and range being finite intervals.

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This graph shows the distance traveled over 5 hours. It is a line segment.

  • The domain represents the time, which goes from 0 to 5 hours. Domain: [0, 5].

  • The range represents the distance, which goes from 0\text{ km} to 50\text{ km}. Range: [0, 50].

  • The y-intercept is (0, 0), representing 0\text{ km} distance at time 0.

  • There is no x-intercept other than the origin, so the zero is x=0.

Examples

Example 5

Consider the graph of the linear function f(x).

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Determine the slope, domain, range, y-intercept, x-intercept, and zero of the function.

Worked Solution
Create a strategy

Calculate the slope using two points. Identify where the line crosses the y-axis (y-intercept) and x-axis (x-intercept). The x-coordinate of the x-intercept is the zero. Determine the domain and range by considering how far the line extends horizontally and vertically.

Apply the idea

  • Slope: The line passes through (0, -4) and (2, 0). The rise is 0 - (-4) = 4 and the run is 2 - 0 = 2. The slope is m = \dfrac{4}{2} = 2.

  • Domain: The line extends infinitely to the left and right. Domain: (-\infty, \infty).

  • Range: The line extends infinitely up and down. Range: (-\infty, \infty).

  • y-intercept: The line crosses the y-axis at -4. The y-intercept is (0, -4).

  • x-intercept: The line crosses the x-axis at 2. The x-intercept is (2, 0).

  • Zero: The y-value is 0 when x=2. The zero is x=2.

Example 6

The graph shows the amount of water remaining in a tank as it drains over 8 minutes.

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Determine the slope (rate of draining), domain, range, y-intercept, and x-intercept (and zero) within the context of the problem.

Worked Solution
Create a strategy

Identify the endpoints of the line segment to calculate the slope (rate). The y-intercept represents the initial amount of water. The x-intercept represents when the tank is empty. The domain represents the time interval, and the range represents the possible amounts of water in the tank.

Apply the idea

  • Slope (Rate): The line segment goes from (0, 100) to (8, 0). The rise is 0 - 100 = -100 liters and the run is 8 - 0 = 8 minutes. The slope is m = \dfrac{-100}{8} = -12.5 liters per minute. The negative sign indicates draining.

  • Domain: The time shown is from 0 to 8 minutes. Domain: [0, 8].

  • Range: The amount of water goes from 100 liters down to 0 liters. Range: [0, 100].

  • y-intercept: The graph starts at (0, 100). This means the initial amount of water was 100 liters.

  • x-intercept: The graph ends at (8, 0). This means the tank is empty after 8 minutes. The x-intercept is (8, 0).

  • Zero: The amount of water is 0 when t=8 minutes. The zero is t=8.

Idea summary

Key features of a linear function's graph include:

  • y-intercept: Where the graph crosses the y-axis (0, y). This is often the initial value.
  • x-intercept: Where the graph crosses the x-axis (x, 0).
  • Zero: The x-value of the x-intercept (where y=0).
  • Domain: All possible x-values. This is often (-\infty, \infty) unless restricted by context.
  • Range: All possible y-values. This is often (-\infty, \infty) (unless the slope is 0) or restricted by context.

These features can usually be identified directly from the graph.

Behavior of linear functions

The slope of a linear function tells us about its behavior, such as whether it is increasing or decreasing and its end behavior.

  • A function is increasing if its y-values increase as its x-values increase (the graph rises from left to right). This occurs when the slope is positive (m > 0).
  • A function is decreasing if its y-values decrease as its x-values increase (the graph falls from left to right). This occurs when the slope is negative (m < 0).
  • A function is constant if its y-values stay the same as its x-values increase (the graph is horizontal). This occurs when the slope is zero (m = 0).
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Increasing (m>0)
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End behavior describes what happens to the y-values of the function as the x-values approach positive infinity (x \to \infty) or negative infinity (x \to -\infty). End behavior describes which way the graph points at the far left and far right ends.

Slope ConditionEnd Behavior
\text{Positive } (m > 0)\text{As } x \to \infty, y \to \infty \text{ (Up to the right)} \\ \text{As } x \to -\infty, y \to -\infty \text{ (Down to the left)}
\text{Negative } (m < 0)\text{As } x \to \infty, y \to -\infty \text{ (Down to the right)} \\ \text{As } x \to -\infty, y \to \infty \text{ (Up to the left)}
\text{Zero } (m = 0)\text{As } x \to \pm\infty, y \to c \text{ (Horizontal)}

An absolute maximum is the highest y-value the function reaches over its entire domain. An absolute minimum is the lowest y-value the function reaches.

Because linear functions with non-zero slopes extend infinitely upwards and downwards, they do not have an absolute maximum or minimum value when their domain is all real numbers (-\infty, \infty). A constant function y=c has both an absolute maximum and an absolute minimum equal to c.

However, if the domain of a linear function is restricted to a closed interval (like [a, b]), the function will have an absolute maximum and an absolute minimum. These occur at the endpoints of the interval.

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This graph shows the height of a balloon between time t=1 and t=7 seconds. The domain is [1, 7].

Since the slope is positive, the function is increasing.

The lowest point is at the left endpoint (1, 5). The absolute minimum height is 5\text{ m}.

The highest point is at the right endpoint (7, 23). The absolute maximum height is 23\text{ m}.

If the slope were negative on a closed interval, the absolute maximum would be at the left endpoint and the absolute minimum at the right endpoint.

Examples

Example 7

Consider the linear function graphed.

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a

Is the function increasing, decreasing, or constant?

Worked Solution
Create a strategy

Determine if the graph rises, falls, or is horizontal from left to right to classify it as increasing, decreasing, or constant.

Apply the idea

The graph falls from left to right. This means the function is decreasing. We can also see the slope is negative (m=-0.5).

b

Describe the end behavior of the function.

Worked Solution
Create a strategy

Observe the direction of the graph as x goes to the far right (\infty) and far left (-\infty) to describe end behavior. Relate these to the slope.

Apply the idea

End Behavior: As x approaches positive infinity (x \to \infty), the graph goes downwards, so y \to -\infty. As x approaches negative infinity (x \to -\infty), the graph goes upwards, so y \to \infty.

Example 8

The graph shows the value of an investment account from year 2 to year 10.

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\text{Year}
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\text{Value (\$)}

What was the absolute minimum and maximum value of the investment during this period?

Worked Solution
Create a strategy

The graph represents the function over a closed interval (from year 2 to year 10). The absolute minimum and maximum values will occur at the endpoints of this interval. Identify the y-values at the endpoints.

Apply the idea

The time interval shown is from year 2 to year 10, so the domain is [2, 10].

The graph starts at the point (2, 5000) and ends at the point (10, 8200).

The function is increasing because it has a positive slope.

  • The lowest value occurs at the left endpoint (t=2). The absolute minimum value is \$5000.

  • The highest value occurs at the right endpoint (t=10). The absolute maximum value is \$8200.

Idea summary

Behavior of Linear Functions:

  • Increasing: Rises left to right (m>0).
  • Decreasing: Falls left to right (m<0).
  • Constant: Horizontal (m=0).
  • End Behavior: Describes y as x \to \pm\infty; depends on the sign of the slope.
  • Absolute Max/Min: Lines that extend infinitely do not have an overall highest or lowest point. On a line segment (a closed interval like [a,b]), the highest and lowest points are at the endpoints.

Comparing linear functions

We often need to compare two or more linear functions. They might be presented in different ways, such as graphs, tables, equations, or verbal descriptions. To compare them, we analyze their key features.

Key features for comparison include:

  • Slope (Rate of Change): Which function is increasing/decreasing faster? Which line is steeper?
  • y-intercept (Initial Value): Which function has a higher or lower starting value?
  • x-intercept (Zero): Which function crosses the x-axis sooner or later?
  • Domain and Range: Are the possible inputs or outputs different, especially in context?
  • Specific Values: How do the function values compare at a particular input?

When comparing functions from graphs, identify these features visually. If comparing a graph to another representation (like an equation or table), first determine the key features from the graph, then determine the same features from the other representation, and finally, compare them using terms like greater than, less than, steeper, faster, higher initial value, etc.

Examples

Example 9

Consider the linear function f(x) that passes through the points A(3,\,5) and B(-2, 10), and the linear function g(x) = 2x - 1.

a

Find the slope of the function f(x).

Worked Solution
Create a strategy

Use the slope formula m= \dfrac{y_2-y_1}{x_2-x_1} with the two given points for f(x). Let point A=(x_1,y_1) and point B=(x_2, y_2).

Apply the idea
\displaystyle m_{f}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the slope formula for f(x)
\displaystyle =\displaystyle \dfrac{10-5}{-2-3}Substitute x_1=3,\,y_1=5,\,x_2=-2,\, and y_2=10
\displaystyle =\displaystyle \dfrac{5}{-5}Evaluate the subtraction
\displaystyle =\displaystyle -1Evaluate the division. The slope of f(x) is -1.
b

Which function, f(x) or g(x), has a greater y-intercept?

Worked Solution
Create a strategy

First, find the equation for f(x) using its slope from part (a) and one of the points in the point-slope form y - y_1 = m(x - x_1).

Then, identify its y-intercept (the value of y when x=0). Identify the y-intercept of g(x) from its equation. Finally, compare the y-intercepts.

Apply the idea

Find the equation for f(x):

\displaystyle y - y_1\displaystyle =\displaystyle m(x - x_1)Write the point-slope form
\displaystyle y - 5\displaystyle =\displaystyle -1(x - 3)Substitute slope m=-1 and point A(3, 5)
\displaystyle y - 5\displaystyle =\displaystyle -x + 3Distribute the -1
\displaystyle y\displaystyle =\displaystyle -x + 8Add 5 to both sides. So, f(x) = -x + 8.

The y-intercept of f(x) is f(0) = -0 + 8 = 8.

The y-intercept of g(x) = 2x - 1 is -1.

Compare the y-intercepts: 8 > -1. Therefore, f(x) has the greater y-intercept.

c

Which function, f(x) or g(x), has a greater value when x = 4?

Worked Solution
Create a strategy

Substitute x = 4 into the equations for f(x) and g(x) and compare the resulting values.

Apply the idea

Calculate f(4):

\displaystyle f(4)\displaystyle =\displaystyle -(4) + 8Substitute x=4 into f(x) = -x + 8
\displaystyle =\displaystyle 4Evaluate

Calculate g(4):

\displaystyle g(4)\displaystyle =\displaystyle 2(4) - 1Substitute x=4 into g(x) = 2x - 1
\displaystyle =\displaystyle 7Evaluate

Comparing the values, 7 > 4. Therefore, g(x) has the greater value when x = 4.

d

Which function, f(x) or g(x), has a greater x-intercept (zero)?

Worked Solution
Create a strategy

Find the x-intercept (zero) for each function by setting the function equal to zero and solving for x. Then, compare the x-values.

Apply the idea

Find the x-intercept for f(x) = -x + 8:

\displaystyle 0\displaystyle =\displaystyle -x + 8Set f(x)=0
\displaystyle x\displaystyle =\displaystyle 8Add x to both sides. The x-intercept is 8.

Find the x-intercept for g(x) = 2x - 1:

\displaystyle 0\displaystyle =\displaystyle 2x - 1Set g(x)=0
\displaystyle 1\displaystyle =\displaystyle 2xAdd 1 to both sides
\displaystyle \dfrac{1}{2}\displaystyle =\displaystyle xDivide both sides by 2. The x-intercept is 0.5.

Comparing the x-intercepts, 8 > 0.5. Therefore, f(x) has the greater x-intercept (zero).

Reflect and check

We calculated the slope and equation for f(x) and used the equation for g(x) to compare their key features like intercepts and specific function values. We can visualize this by graphing both functions.

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x
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y

The graph confirms that f(x) has a negative slope, a higher y-intercept, and a greater x-intercept than g(x). It also shows that at x=4, the dashed line (g(x)) is higher than the solid line (f(x)).

Example 10

Compare the two linear functions, f(x) (solid line) and g(x) (dashed line), shown on the graph.

x
y
a

Which function has a greater slope?

Worked Solution
Create a strategy

Determine the slope for each function from the graph and then compare the values.

Apply the idea

For function f(x) (solid line), the slope can be found using points (0, 1) and (1, 3). The slope is m = 2.

For function g(x) (dashed line), the slope can be found using points (0, 4) and (4, 0). The slope is m = -1.

Since 2 > -1, function f(x) has the greater slope.

b

Which function has a greater y-intercept?

Worked Solution
Create a strategy

Determine the y-intercept for each function directly from the graph, then compare the values.

Apply the idea

From the graph, the y-intercept of f(x) is (0, 1).

The y-intercept of g(x) is (0, 4).

Since 4 > 1, function g(x) has a greater y-intercept.

c

Which function has a greater x-intercept (zero)?

Worked Solution
Create a strategy

Determine the x-intercept for each function directly from the graph. Then compare the values.

Apply the idea

From the graph, the x-intercept of f(x) is (-0.5, 0). Its zero is x=-0.5.

The x-intercept of g(x) is (4, 0). Its zero is x=4.

Since 4 > -0.5, function g(x) has a greater x-intercept (zero).

Example 11

Two companies offer streaming plans. Company A's plan is shown in the graph. Company B charges a \$5 sign-up fee plus \$10 per month.

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\text{Months}
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\text{Total Cost (\$)}

Which company has a higher monthly rate? Which company has a higher cost for the first month?

Worked Solution
Create a strategy

Determine the slope (monthly rate) and y-intercept (initial fee) for Company A from the graph.

Determine the same features for Company B from the verbal description. Compare the rates and the costs at month 1.

Apply the idea

For Company A (from the graph):

  • Slope (Monthly Rate): Passes through (0, 0) and (5, 60). Rate =\dfrac{60 - 0}{5 - 0} = \$12 per month.
  • y-intercept (Initial Fee): Crosses the y-axis at 0. The initial fee is \$0.
  • Cost for first month (x=1): 12(1) = \$12.

For Company B (from the description):

  • Slope (Monthly Rate): Given as \$10 per month.
  • y-intercept (Initial Fee): The sign-up fee is \$5.
  • Cost for first month (x=1): \$5 \text{(fee)} + \$10 \text{(first month)} = \$15.

Comparing the two companies:

  • Monthly Rate: Company A's rate (\$12/month) is higher than Company B's rate (\$10/month).
  • Cost for First Month: Company B's cost (\$15) is higher than Company A's cost (\$12).
Idea summary

To compare linear functions, we analyze their key features:

  • Slope: Compare rates or steepness.
  • y-intercept: Compare initial values.
  • x-intercept/Zero: Compare where the functions equal zero.
  • Domain/Range: Compare constraints, especially in context.
  • Specific Values: Compare outputs for a given input.

Extract these features from the given representations (graph, table, equation, verbal) before making comparisons.

Outcomes

AFDA.AF.1g

Compare and contrast linear, quadratic, and exponential functions using multiple representations (e.g., graphs, tables, equations, verbal descriptions).

AFDA.AF.2a

Determine the domain and range of a function given a graphical representation, including those limited by contexts.

AFDA.AF.2b

Identify intervals on a graph for which a function is increasing, decreasing, or constant.

AFDA.AF.2c

Given a graph, identify the location and value of the absolute maximum and absolute minimum of a function over the domain of a function.

AFDA.AF.2d

Given a graph, determine the zeros and intercepts of a function.

AFDA.AF.2e

Describe and recognize the connection between points on the graph and the value of a function.

AFDA.AF.2f

Describe the end behavior of a function given its graph.

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