Question

Graph the function. $$ g(x)=-3 x-2 $$

Short answer

The graph of the function \(g(x) = -3x - 2\) is a straight line with a slope of -3 and crossing the y-axis at -2.

Step-by-step solution

Identifying the Slope and Y-Intercept

From the given function \(g(x) = -3x - 2\), it can be identified that the slope \(m = -3\) and the y-intercept \(c = -2\). These are obtained from the general linear equation form \(y = mx + c\).

Plotting the Y-Intercept

The y-intercept (\(c\)) is the point where the line crosses the y-axis. This happens when \(x = 0\). So for this function, when \(x = 0\), \(y = -2\). So we plot the point (0, -2) as the y-intercept.

Drawing the Line using the Slope

The slope (\(m\)) of -3 means that for every unit increase in \(x\), \(y\) decreases by 3 units. From the y-intercept, move one step to the right and three steps downward. Draw a line through these two points to obtain the graph of the function.

Key concepts

Slope of a Line

Understanding the slope of a line is crucial for graphing linear functions. The slope is a measure of how steep the line is and indicates the rate of change from one point to another on the line. It is usually represented by the letter 'm' in the slope-intercept form of a linear equation, which is \(y = mx + c\).

If you have the equation \(g(x) = -3x - 2\), the slope here is \(m = -3\). This tells us that for each step you move to the right along the x-axis, the value of \(g(x)\), or \(y\), goes down by three steps. It's like walking down a hill: if the slope is negative, you're heading downward. If the slope were positive, you would go up as you move right. A steeper slope means a faster increase or decrease as you move along the line.

• A positive slope travels uphill from left to right.
• A negative slope travels downhill from left to right.
• A zero slope is a flat line, indicating no change.
• An undefined slope represents a vertical line, indicating infinite steepness.

Using the slope, we can predict the behavior of the line without even drawing it, providing insight into the relationship between variables in an equation.

Y-Intercept

The y-intercept is another fundamental aspect in the equation of a line and is signified by the letter 'c'. It's where the line crosses the y-axis. Specifically, it happens when the input \(x = 0\). The y-intercept is a snapshot of where the line starts from or passes through on the y-axis.

For our example, \(g(x) = -3x - 2\), the y-intercept is at \(c = -2\). This is the starting point for graphing the line. Simply put, it's the launching pad for the line on the graph. To visualize it, find the y-axis on your graph, go straight to the value of \(c\), and put a dot there. For a positive \(c\), go up from the origin; for a negative \(c\), go down.

The y-intercept offers valuable information:

• It gives a line's location relative to the origin.
• It provides a clear starting point for graph construction.
• It shows the output value when input is zero.

Knowing the y-intercept allows us to draw the line accurately and interpret the function it represents.

Linear Equation

A linear equation forms the foundation of a straight line graph, and it can be represented in multiple ways, with the most common being the slope-intercept form: \(y = mx + c\). The two components we've discussed, the slope 'm' and y-intercept 'c', come together to create this equation.

In our function \(g(x) = -3x - 2\), we're looking at a linear equation in terms of 'g', a variable just like 'y'. This equation tells us exactly how 'g' (or 'y') will change with every change in 'x'. It's a simple, yet powerful, expression of a relationship between two variables.

Here's the big picture of what a linear equation tells you:

• The slope indicates the rate of change between the variables.
• The y-intercept provides a starting point on the y-axis.
• The equation is a rule that tells us how to find a 'y' for each 'x'.
• Using the equation, you can calculate the coordinates of any point on the line.

Linear equations are everywhere in real life, from calculating simple budgets to understanding complex scientific relationships. Grasping how to work with linear equations is essential for solving various mathematical problems and interpreting real-world scenarios.