Mathematics: Functions – Grade 8

Understanding functions begins with recognizing the fundamental property that distinguishes them from other mathematical relationships. This property is both simple and profound: a function assigns exactly one output value to each input value.

Imagine you're organizing a school dance where each student must be paired with exactly one chaperone. If every student has exactly one chaperone assigned to them, this represents a functional relationship. However, if some students have multiple chaperones or no chaperone at all, the relationship would not be functional.

In mathematical terms, we call the input values the domain and the output values the range. The input variable is also called the independent variable because you can choose its value independently. The output variable is the dependent variable because its value depends on what you choose for the input.

For example, consider the relationship between the number of hours you work and the money you earn at $\$12$ per hour. If $h$ represents hours worked and $M$ represents money earned, then $M = 12h$. For each number of hours (input), there's exactly one amount of money earned (output), making this a function.

Ordered Pairs and Sets

When given a set of ordered pairs like ${(3, 4), (-2, 3), (7, 1), (-\frac{1}{2}, 4), (-2, \frac{1}{2})}$, examine whether any input value (first coordinate) appears more than once with different output values (second coordinates). In this example, the input $-2$ appears twice with different outputs ($3$ and $\frac{1}{2}$), so this relationship is not a function.

Tables

When examining tables, check if any x-value appears multiple times with different y-values. If each x-value has exactly one corresponding y-value, the table represents a function. Remember that the same y-value can correspond to different x-values—this doesn't violate the function property.

Graphs and the Vertical Line Test

For graphs, use the vertical line test: if you can draw any vertical line that intersects the graph more than once, the graph does not represent a function. This test works because a vertical line represents all points with the same x-coordinate (input), and if it hits the graph multiple times, that means one input has multiple outputs.

Mapping Diagrams

Mapping diagrams show input values on one side connected by arrows to output values on the other side. For a function, each input should have exactly one arrow coming from it, though multiple inputs can point to the same output.

The domain includes all possible input values for which the function is defined. The range includes all possible output values that the function can produce.

Expressing Domain and Range

You can express domain and range in several ways:

1. List notation: For discrete sets like ${4, 5, 6, 7, 8}$
2. Inequality notation: For continuous intervals like $x ≥ 0$ or $-3 ≤ y ≤ 15$
3. Verbal descriptions: "All real numbers greater than or equal to zero"

The choice depends on the context and type of relationship. For a discrete set of ordered pairs, use list notation. For continuous functions representing real-world situations, inequalities or verbal descriptions often work better.

Consider a cell phone plan that charges $\$25$ per month plus $\$0.10$ per text message. If $t$ represents the number of texts and $C$ represents the total cost, then $C = 25 + 0.10t$.

• Domain: All non-negative integers (you can't send a negative number of texts)
• Range: All values $C ≥ 25$ (the minimum cost is $\$25$ even with zero texts)

This relationship is a function because each number of text messages corresponds to exactly one total cost.

Students sometimes confuse independent and dependent variables. Remember: the independent variable is the input you control (like hours worked), while the dependent variable is the output that depends on your input (like money earned). Think of cause and effect—the independent variable is the cause, and the dependent variable is the effect.

Another common error is thinking that if two different inputs produce the same output, the relationship isn't a function. This is incorrect! Multiple students can have the same grade on a test, and it's still a functional relationship from student names to grades. The restriction is that each input can have only one output, not that each output must come from only one input.

Linear functions represent the simplest and most fundamental type of function, characterized by their constant rate of change. Understanding how to identify linear functions from various representations is crucial for success in algebra and beyond.

A linear function has a constant rate of change between its input and output values. This means that for equal increases in the input variable, the output variable always changes by the same amount. Graphically, linear functions appear as straight lines, which is why they're called "linear."

Think about earning money at a constant hourly rate of $\$15$ per hour. If you work 1 hour, you earn $\$15$. If you work 2 hours, you earn $\$30$. Each additional hour worked increases your earnings by exactly $\$15$—this constant rate of change makes the relationship linear.

Linear functions can be written in the general form $y = mx + b$, where:

• $m$ is the slope (rate of change)
• $b$ is the y-intercept (output value when input is zero)
• $x$ is the independent variable
• $y$ is the dependent variable

Examples of linear function equations:

• $y = 3x + 2$ (slope = 3, y-intercept = 2)
• $y = -0.5x + 7$ (slope = -0.5, y-intercept = 7)
• $y = 4x$ (slope = 4, y-intercept = 0)
• $y = -2$ (slope = 0, y-intercept = -2, this is a horizontal line)

Non-linear equations include:

• $y = x^2 + 3$ (contains $x^2$)
• $y = \frac{1}{x}$ ($x$ is in the denominator)
• $y = 2^x$ ($x$ is an exponent)

When examining a table to determine if it represents a linear function, calculate the rate of change between consecutive points. For a linear function, this rate should remain constant.

Method for Tables:

1. Check that x-values increase by equal amounts
2. Calculate the change in y-values for each equal x-interval
3. If all y-changes are equal, the function is linear

For example, consider this table:

As $x$ increases by 1, $y$ increases by 3 each time. The constant rate of change of 3 confirms this is a linear function with equation $y = 3x + 2$.

Contrast this with a non-linear table:

Here, $y$ increases by 3, then 5, then 7—the rate of change is not constant, indicating a non-linear function (specifically $y = x^2$).

Linear function graphs are straight lines. Any graph that curves, bends, or has varying steepness represents a non-linear function.

Key visual characteristics of linear functions:

• Perfectly straight line (no curves or bends)
• Constant steepness throughout
• Extends infinitely in both directions (unless domain is restricted)
• Can have positive slope (rising), negative slope (falling), or zero slope (horizontal)

A powerful way to understand linear functions is to create both a table and graph for the same relationship. Consider the isosceles right triangle area function $A = 0.5s^2$, where $A$ is the area and $s$ is the length of the legs.

Creating a table:

Plotting these points reveals a curved graph, not a straight line. The rate of change increases (0.5 to 1.5 to 2.5), confirming this is not a linear function.

It's important to understand that not all linear relationships are proportional, but all proportional relationships are linear.

Proportional relationships:

• Pass through the origin (0, 0)
• Have the form $y = kx$ (no y-intercept term)
• Examples: $y = 3x$, $y = -0.5x$

Linear but not proportional:

• Do not pass through the origin
• Have the form $y = mx + b$ where $b ≠ 0$
• Examples: $y = 2x + 5$, $y = -x + 3$

Linear functions model many real-world situations:

Temperature Conversion: $F = \frac{9}{5}C + 32$ (Celsius to Fahrenheit) Car Rental: $C = 25 + 0.15m$ ($\$25$ base fee plus $\$0.15$ per mile) Gym Membership: $C = 50 + 20m$ ($\$50$ signup fee plus $\$20$ per month)

Each of these has a constant rate of change, making them linear functions.

When determining if a function is linear:

1. From equations: Look for the form $y = mx + b$
2. From tables: Calculate rates of change between consecutive points
3. From graphs: Check if the graph is a straight line
4. From descriptions: Identify if there's a constant rate of change

Remember that context matters. A table might not represent a function at all if x-values repeat with different y-values, so always check the function property first before determining linearity.

Understanding how functions behave over different intervals provides valuable insights into real-world relationships and mathematical patterns. By analyzing when functions increase, decrease, or remain constant, you can interpret data trends, predict future values, and make informed decisions based on mathematical models.

Function behavior describes how the output values change as the input values increase. There are three basic types of behavior:

Increasing: As $x$ increases, $y$ increases. The function is "rising" or "going up." Decreasing: As $x$ increases, $y$ decreases. The function is "falling" or "going down." Constant: As $x$ increases, $y$ stays the same. The function is "flat" or "horizontal."

A single function can exhibit different behaviors over different intervals of its domain. For example, a function might increase from $x = 0$ to $x = 3$, then decrease from $x = 3$ to $x = 7$, and finally remain constant from $x = 7$ onwards.

Graphs provide the most visual way to analyze function behavior. When reading from left to right (increasing $x$-values):

Increasing intervals: The graph slopes upward ↗️ Decreasing intervals: The graph slopes downward ↘️ Constant intervals: The graph is horizontal →

Consider a graph showing the number of bacteria in a culture over time. The graph might show:

• Hours 0-1: Constant (no growth in bacterial count)
• Hours 1-3: Increasing rapidly (exponential growth phase)
• Hours 3-4: Constant again (growth stops as nutrients deplete)
• Hours 4-8: Decreasing (bacteria die off as waste accumulates)

Real-world scenarios often describe function behavior through words rather than graphs or equations. Learning to translate these descriptions into mathematical understanding is a crucial skill.

Example: Madison's Bacterial Growth Study

Madison observes four phases of bacterial growth:

• Phase 1 (0-1 hours): "No growth in the number of cells" → Constant
• Phase 2 (1-3 hours): "Rapid growth in the number of bacteria" → Increasing
• Phase 3 (3-4 hours): "Growth stops as nutrients are used up" → Constant
• Phase 4 (4-8 hours): "All bacteria gradually die off" → Decreasing

From this description, you can sketch a graph that starts flat, rises steeply, becomes flat again, then falls gradually.

When describing function behavior, it's essential to specify the intervals over which the behavior occurs. Use proper interval notation or clear verbal descriptions:

Correct: "The function is increasing on the interval from $x = 2$ to $x = 5$." Incorrect: "The function is increasing at $x = 2$." (This describes behavior at a single point, not an interval)

Remember that domain and range can affect your analysis. If a function represents a real-world situation like "number of students in a classroom over time," negative values for either variable might not make sense in the context.

Temperature Throughout a Day

Consider the temperature in your city from midnight to midnight:

• 12 AM - 6 AM: Decreasing (coolest part of night)
• 6 AM - 2 PM: Increasing (sun rises and heats the day)
• 2 PM - 6 PM: Constant or slightly decreasing (peak heat, then cooling begins)
• 6 PM - 12 AM: Decreasing (evening cooldown)

Stock Market Analysis

A stock price over a trading day might show:

• Opening: Constant (pre-market, no trading)
• Morning: Increasing (positive news drives price up)
• Midday: Decreasing (profit-taking causes decline)
• Afternoon: Increasing again (renewed buying interest)

When asked to sketch a graph from a written description, focus on the overall trends rather than exact mathematical precision. You're demonstrating understanding of function behavior, not creating precise mathematical models.

Guidelines for Sketching:

1. Start by identifying the phases and their behaviors
2. Decide on appropriate scales for both axes
3. Use straight line segments or gentle curves as appropriate
4. Label axes clearly with units
5. Ensure your graph reflects all described behaviors

For rapid increases, use steeper slopes. For gradual changes, use gentler slopes. For constant behavior, draw horizontal line segments.

Function behavior analysis connects to several important mathematical ideas:

Slope: In linear functions, positive slope indicates increasing behavior, negative slope indicates decreasing behavior, and zero slope indicates constant behavior.

Rate of Change: The steepness of increase or decrease tells you about the rate of change. Steeper slopes indicate faster rates of change.

Optimization: Finding where functions change from increasing to decreasing (or vice versa) helps identify maximum and minimum values—concepts you'll explore further in advanced mathematics.

Students sometimes confuse the $x$ and $y$ axes when describing behavior. Remember:

• Increasing/Decreasing refers to what happens to the $y$-values (outputs) as $x$-values (inputs) increase
• Domain intervals specify the range of $x$-values you're analyzing
• A function can have the same $y$-value at different $x$-values without being constant (constant means $y$ doesn't change over an interval of $x$-values)

Understanding function behavior helps in:

• Science: Analyzing experimental data and identifying trends
• Economics: Interpreting market trends and economic indicators
• Sports: Analyzing performance data over time
• Environmental Science: Understanding population changes, pollution levels, or climate data
• Personal Finance: Tracking savings growth or debt reduction over time

By mastering function behavior analysis, you develop critical thinking skills that apply far beyond mathematics class, helping you interpret data and make informed decisions in many areas of life 📈.