91Ó°ÊÓ

Understanding linear equations is essential for graphing functions. A linear equation is any equation that can be written in the form: \ y = mx + b \, where \( m \) represents the slope, and \( b \) represents the y-intercept. In our case, the function is \( f(x) = -3x \). This is still a linear equation, but it has no y-intercept term (or \( b \)), meaning it can also be written as \( y = -3x + 0 \).
Linear equations always produce straight lines when graphed because the rate of change between \( x \) and \( y \) is constant. This rate of change is called the slope.

• The slope \( m \) in the equation \( f(x) = -3x \) is \( -3 \).
• A positive slope means the line rises as you move from left to right.
• A negative slope, like here, means the line falls as you move from left to right.

Creating a table of values helps us see this constant rate of change clearly. By plotting points and connecting them with a straight line, we visualize the linear equation.

Next, let's explore the domain and range of the function \( f(x) = -3x \).

**Domain:**
The domain is all possible input values (\( x \)) for which the function is defined. For linear functions, there are no restrictions on \( x \). You can plug in any real number, and the function will output a corresponding \( y \) value. Therefore, the domain of \( f(x) = -3x \) is \ ( -\infty, \infty ) \ which means all real numbers.
**Range:**
The range is all possible output values (\( y \)) the function can produce. For \( f(x) = -3x \), as \( x \) takes any real number value, \( y \) will also take any real number value. This is because the multiplication of any real number with \( -3 \) can yield any real number. Thus, the range of this function is also \ ( -\infty, \infty ) \.
Both the domain and range being all real numbers highlights an important property of linear functions without restriction.

To graph a function, you need to plot it on a coordinate plane. This is a two-dimensional plane formed by the intersection of a horizontal line called the x-axis and a vertical line called the y-axis. Each point is represented by a pair of coordinates \( (x, y) \).
To graph \( f(x) = -3x \), we plot points based on our table of values;

• For example, when \( x = -2 \), \( y = -3(-2) = 6 \), so we plot the point (-2, 6).
• When \( x = 0 \), \( y = 0 \) points us to (0, 0).
• When \( x = 2 \), \( y = -3(2) = -6 \), thus we plot (2, -6).

After plotting these points, we connect them with a straight line. This line extends infinitely in both directions because the domain and range of \( f(x) = -3x \) are both all real numbers.
Remember, the coordinate plane allows us to visually represent the relationship between \( x \) and \( y \) in the function. Each point signifies an exact solution to the equation.