Problem 83 - Algebra A Combined Function

Question

Graph each linear function. \(f(x)=-3 x\)

Step-by-Step Solution

Answer

Graph a line through the origin with a slope of -3, plotting points like (0,0), (1,-3), and (-1,3).

Step 1: Understanding the Function

The function given is a linear function in the form of \(f(x) = -3x\). This represents a straight line through the origin with a slope of -3.

Step 2: Identify Key Features

Identify the y-intercept and the slope. For the function \(f(x) = -3x\), the y-intercept is 0 (the line passes through the origin) and the slope is -3. This means for every 1 unit increase in \(x\), \(f(x)\) decreases by 3 units.

Step 3: Plot Points

Start plotting the graph by identifying easy points. At \(x = 0\), \(f(x) = 0\); this is our first point (0,0). At \(x = 1\), \(f(x) = -3(1) = -3\); plot point (1, -3). At \(x = -1\), \(f(x) = -3(-1) = 3\); plot point (-1, 3).

Step 4: Draw the Line

Using the points plotted, draw a straight line through them. Make sure the line extends across the graph (through both the positive and negative sides of the x-axis) since the function is defined for all real numbers \(x\).

Key Concepts

Slope of a Line

In a linear function, the slope is a crucial characteristic that determines how the line tilts. The slope is often represented by the letter \(m\). It tells us how steep the line is and what its direction is. In mathematical terms, the slope is calculated as the "rise" over the "run" or the change in \(y\) over the change in \(x\).
For example, in the linear function \(f(x) = -3x\), the slope \(m\) is \(-3\). This means that for every unit increase along the \(x\)-axis, the \(y\)-value decreases by 3 units. A negative slope number indicates that the line is descending from left to right.
Beyond acting as a guideline for drawing a line, the slope is vital for understanding the relationship between variables. If the slope was positive, the function would increase as \(x\) increases. A zero slope implies a horizontal line, while an undefined slope (division by zero) describes a vertical line.

Y-Intercept

The y-intercept of a line is the exact point where the line crosses the y-axis. This spot is important as it provides a starting point for graphing the line, especially if you know the slope already.
In the function \(f(x) = -3x\), the y-intercept is 0. We find this by identifying the constant term when the equation is in the form \(y = mx + b\). In our case, the expression can be viewed as \(y = -3x + 0\), making the y-intercept \(0\), indicating that the line passes through the origin.
Understanding the y-intercept can help us make real-world connections. For instance, if this function represented a cost model, the y-intercept would indicate the starting or fixed cost before any additional charges based on the slope (or rate of change) are applied.

Plotting Points

Plotting points is the practical step of graphing a linear function. It involves marking specific coordinates on the graph that satisfy the equation. Once you plot multiple points that align with the function, you connect them with a straight line to represent the linear function.
To graph \(f(x) = -3x\), start by choosing values for \(x\) to easily calculate the corresponding \(y\) values:

• When \(x = 0\), \(f(x) = 0\), giving the point (0, 0).
• With \(x = 1\), \(f(x) = -3(1) = -3\), resulting in the point (1, -3).
• If \(x = -1\), \(f(x) = -3(-1) = 3\), creating the point (-1, 3).

By plotting these points and drawing a line through them, you create a visual of the function's behavior. The line can extend continuously across the graph, demonstrating how the function applies for every real number \(x\).
It's important to verify that all plotted points align straight, ensuring the line accurately represents the function.