Problem 37 - Algebra 1

Question

GRAPHING FUNCTIONS Graph the function. $$f(x)=3 x-9$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 3x - 9\) is a straight line that slopes upward from left to right, starting from the y-intercept point (-9).

Step 1: Identify the Slope and y-intercept

In our function \(f(x) = 3x - 9\), the coefficient of x is the slope of the function. Hence, the slope of the function is 3. The constant term in the function is our y-intercept, so the y-intercept is -9.

Step 2: Draw the y-intercept

Start by making a mark at the point (0, -9) on the graph which is our y-intercept.

Step 3: Use the Slope to Determine the Next Point

From the y-intercept, the slope of the line is rise over run. Our slope is 3 which we can consider as 3/1, so we rise up 3 units and run 1 unit to the right from our y-intercept to draw our line.

Step 4: Draw the Line

Using a straight edge or ruler, draw a line that passes through the points located at (0, -9) and (1, -6). The resultant line is the graph of the function \(f(x) = 3x - 9\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to represent a linear equation. This form is given by the equation \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This form is particularly useful because it allows you to quickly identify key features of a linear function.
It's like having a roadmap for drawing the graph of a linear equation.

• \(m\) = Slope: This tells you how steep the graph is.
• \(b\) = Y-intercept: This shows where the line will cross the y-axis.

For example, consider the equation \(f(x) = 3x - 9\). Here, the slope \(m\) is 3 and the y-intercept \(b\) is -9. This tells you that for each step to the right (along the x-axis), the line goes up three steps (rise).
And the graph will intersect the y-axis at -9.

Y-Intercept

The y-intercept is the point where the graph of a function crosses the y-axis. This is a crucial point because it provides a starting point for drawing the graph of a linear equation.
When you have a linear function in the slope-intercept form \( y = mx + b \), the y-intercept \(b\) is the constant term.

• To find it, simply look at the number that isn't attached to \(x\) in the equation.
• In \(f(x) = 3x - 9\), the y-intercept is -9, represented by the point (0, -9).

From the y-intercept, you can use the slope to find other points on the line.
By identifying the y-intercept, you make graphing the entire linear function much easier.

Linear Equations

Linear equations are a type of algebraic equation where each variable is raised to the power of one. These equations graph as straight lines when plotted on a coordinate plane.
They are fundamental in algebra and provide a simple way to model relationships between variables.

• General form: \(ax + by = c\)
• Slope-intercept form: \(y = mx + b\)

Understanding how these linear equations work can help in solving and graphing them effectively.
Let's consider \(f(x) = 3x - 9\).

• This is a linear equation because it forms a straight line on a graph.
• The equation tells you that for every increase of 1 unit in \(x\), \(f(x)\) increases by 3 units because the slope is 3.

Linear equations often appear in both real-world and theoretical contexts, making them essential in math learning.