The Main Idea 

A linear function is characterized by a constant rate of change and can be represented as a polynomial of degree [latex]1[/latex].

The Main Idea 

The Language of Linear Functions

• Word Form: Describing a linear function in words helps you understand the relationship between variables. For instance, if a train’s distance from a station is determined by its speed and initial distance, you can say, “The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station.”
• Rate of Change: This is another term for the slope. It tells you how fast one variable changes in relation to another.

Understanding the language used to describe linear functions can be a game-changer. For example, the term “rate of change” might sound complex, but it’s just a fancy way of saying how one variable (like time) affects another (like distance). In real-world scenarios like the train example, this rate is constant, making it easier to predict future outcomes.

Function Notation and Its Importance

• Slope-Intercept Form: This is the equation [latex]y=mx+b[/latex], where [latex]m[/latex] is the slope and [latex]b[/latex] is the [latex]y[/latex]-intercept. It’s a standardized way to write linear functions.
• Function Notation: [latex]f(x)=mx+b[/latex] is another way to write the slope-intercept form, emphasizing that [latex]y[/latex] is a function of [latex]x[/latex].

Function notation isn’t just for mathematicians; it’s a useful tool for anyone wanting to understand relationships between variables. For example, [latex]D(t)=83t+250[/latex] tells us that the distance [latex]D[/latex] is a function of time [latex]t[/latex]. This notation makes it clear what the input and output variables are, which is crucial for understanding the function’s behavior.

Visualizing Linear Functions

• Tabular Form: A table can show how the output changes with each unit increase in the input.
• Graphical Form: A graph provides a visual representation of a linear function. The slope and [latex]y[/latex]-intercept can be easily identified.

Graphs offer a powerful way to visualize linear functions. For instance, the graph of [latex]D(t)=83t+250[/latex] would be a straight line, allowing you to instantly see the rate of change and initial value. This can be particularly helpful in real-world applications like determining how far a train will be from a station at a given time.

The Main Idea 

Increasing Linear Function: If you’re looking at a graph and see the line slanting upward from left to right, you’re likely dealing with an increasing function.

Decreasing Linear Function: A line that slants downward from left to right on a graph usually signifies a decreasing function.

Constant Linear Function: A horizontal line on a graph is a dead giveaway for a constant function.

Graphing is a powerful tool for understanding linear functions. Use online graphing calculators like Desmos to visualize these functions. You can even add sliders to manipulate the slope. For instance, the function [latex]f(x)=−\frac{2}{32}x−\frac{4}{3}[/latex] can be graphed to visually represent its decreasing nature.

The Main Idea 

Slope is the measure of how steep a line is.

It’s calculated as the change in output (rise) divided by the change in input (run).

The formula to calculate slope is [latex]m=\dfrac{\text{change in output (rise)}}{\text{change in input (run)}}=\dfrac{\Delta y}{\Delta x}=\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \Rightarrow \dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[/latex]

Slope isn’t just a mathematical concept; it’s a real-world indicator of change. Think of it as the speedometer of your car. A high absolute value of the slope means you’re either accelerating fast or decelerating quickly—essentially, you’re going through a lot of change. A slope close to zero? You’re cruising at a steady pace.

Quick Tips:

• Units for slope are always [latex]\frac{\text{units for the output}}{\text{units for the input}}[/latex]
• When you’re calculating slope, don’t forget about the units. They give context to your numbers. For instance, if you’re looking at a graph that represents the speed of a car over time, the slope could be in “miles per hour.” This tells you how fast the car is going, which is much more informative than just a number.

The Main Idea 

The [latex]x[/latex]-intercept is the point where a line crosses the [latex]x[/latex]-axis. In mathematical terms, it’s the value of [latex]x[/latex] when [latex]f(x) = 0[/latex]. For example, for the function [latex]f(x) = 3x - 6[/latex], the [latex]x[/latex]-intercept is found by setting [latex]f(x)[/latex] equal to zero and solving for [latex]x[/latex].

Set the Function to Zero: To find the [latex]x[/latex]-intercept, set [latex]f(x)=0[/latex] and solve for [latex]x[/latex].

Check for Exceptions: Not all linear functions have [latex]x[/latex]-intercepts. Functions of the form [latex]y=c[/latex], where [latex]c[/latex] is a nonzero real number, don’t have an [latex]x[/latex]-intercept.