Problem 27 - Algebra Form and Function

Question

Identify the slope and \(y\) -intercept and graph the function. $$ f(x)=3 x-2 $$

Step-by-Step Solution

Answer

Answer: The slope of the function is \(3\) and the y-intercept is \(-2\). To graph it, plot the y-intercept point \((0, -2)\) on the coordinate plane. Then, use the slope as a ratio \(\frac{3}{1}\) to find another point by moving up 3 units and 1 unit to the right from the y-intercept. Plot this point \((1,1)\) and draw a straight line through both points. This line represents the graph of the function \(f(x) = 3x - 2\).

Step 1: Identify the slope

Compare the given function \(f(x) = 3x -2\) with the standard linear equation form \(f(x) = mx + b\). We can see that \(m = 3\). So, the slope of the function is \(3\).

Step 2: Identify the y-intercept

Comparing the given equation with the standard linear equation form, we can see that the y-intercept \(b=-2\).

Step 3: Set up the graph

To graph the function \(f(x) = 3x - 2\), first, set up a coordinate plane with labeled \(x\) and \(y\) axes. Start by plotting the y-intercept, which is the point \((0, -2)\) on the graph.

Step 4: Use the slope to plot other points

The slope of the function is \(3\). The slope can be rewritten as a ratio, \(\frac{3}{1}\), representing the rise over the run. Starting from the y-intercept point \((0, -2)\), go up \(3\) units (rise) and run \(1\) unit to the right (run). This will give you another point on the graph at \((1,1)\).

Step 5: Draw the line

Now that we have two points \((0, -2)\) and \((1,1)\), draw a straight line that passes through these points. This line represents the graph of the function \(f(x) = 3x - 2\).

Key Concepts

Slope

The slope of a line is a measure of how steep the line is. In a linear equation in the form of \(f(x) = mx + b\), the coefficient \(m\) represents the slope. The slope tells us how much \(y\) changes for every change in \(x\).

The slope can be thought of as "rise over run," which means how much the line goes up (rise) for each step it moves horizontally (run). For example, in the equation \(f(x) = 3x - 2\), the slope \(m\) is 3. This tells us that for every unit the line moves right on the x-axis, it moves up 3 units on the y-axis.

Interpreting slopes can give us valuable insights:

• A positive slope means the line is rising as it moves from left to right.
• A negative slope means the line is descending.
• A zero slope means the line is horizontal.
• An undefined slope (division by zero) indicates a vertical line.

Understanding slope is key in graphing because it helps us determine the direction and steepness of the line.

Y-intercept

The \(y\)-intercept of a line is where the line crosses the \(y\)-axis on a graph. In the equation \(f(x) = mx + b\), the constant term \(b\) represents the \(y\)-intercept. This is where \(x\) equals zero.

So, when you substitute \(x = 0\) into the equation, \(f(x) = b\), which gives you the \(y\)-intercept directly. In our example equation \(f(x) = 3x - 2\), the \(y\)-intercept is -2. This means the line crosses the \(y\)-axis at the point \((0, -2)\).

The \(y\)-intercept helps with:

• Starting point for graphing the line.
• Understanding where the function will intersect the \(y\)-axis.

A good practice is to always begin by plotting the \(y\)-intercept before applying the slope to find additional points.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations and visualize the relationships between variables. It consists of a horizontal line, called the \(x\)-axis, and a vertical line, called the \(y\)-axis, intersecting at a point called the origin \((0,0)\).

Each point on the plane is defined by a pair of numbers \((x, y)\), known as coordinates. This is crucial for graphing linear equations like \(f(x) = 3x - 2\). The \(x\)-coordinate represents the horizontal position, while the \(y\)-coordinate represents the vertical position.

Graphing on the coordinate plane involves several steps:

• First, plot the \(y\)-intercept where the line crosses the \(y\)-axis.
• Then, use the slope to find another point by "rising" and "running" from the intercept.
• Finally, draw a line through the points.

The coordinate plane helps us to easily see how equations like \(f(x) = 3x - 2\) translate into visual lines.