Problem 10 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=3+2 x $$

Step-by-Step Solution

Answer

Graph a line through points (0, 3), (1, 5), and (-1, 1) for \(f(x) = 3+2x\).

Step 1: Identify Key Characteristics

The given function is a linear function, written in the general form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this function, \(m = 2\) and \(b = 3\). This means the graph is a straight line with a slope of 2 and a y-intercept of 3.

Step 2: Plot the Y-Intercept

Start by plotting the y-intercept of the function on the graph. Here, the y-intercept is the point where the graph crosses the y-axis. For \(f(x) = 3 + 2x\), the y-intercept is 3. Plot the point (0, 3) on the graph.

Step 3: Calculate Additional Points

To determine the shape of the graph, calculate a few more points. Choose different values for \(x\) and calculate the corresponding \(y\) using the function \(f(x) = 3 + 2x\). For \(x = 1\), \(f(1) = 3 + 2(1) = 5\), so plot (1, 5). For \(x = -1\), \(f(-1) = 3 + 2(-1) = 1\), so plot (-1, 1).

Step 4: Draw the Line

With at least two points plotted, extend a straight line through the points (0, 3), (1, 5), and (-1, 1). Ensure the line continues in both directions, as a linear function extends infinitely.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation, helping you understand the graph of a line quickly and easily.
It takes the format \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

In our exercise, you can see that our linear equation \( f(x) = 3 + 2x \) is clearly in slope-intercept form. We find that:

• The slope \( m = 2 \), meaning the line rises 2 units up for every 1 unit it moves to the right.
• The y-intercept \( b = 3 \), indicating that the line crosses the y-axis at the point (0, 3).

Understanding the slope-intercept form gives you a quick snapshot of the line's steepness and where it starts on the graph. This makes it straightforward to draw and analyze linear graphs.
Whenever you need to graph a linear equation, translating it into slope-intercept form is often your first step.

Plotting Points

To accurately sketch the graph of a linear equation, start by plotting points. This method helps visualize the line you're about to draw.
Begin with the y-intercept due to its prominence in the slope-intercept form. For our example, the y-intercept is 3, so you would start by placing a point at (0, 3) on the graph.
Next, calculate more points using other values for \( x \). You simply substitute these values into the equation to find their corresponding \( y \) values:

• Choosing \( x = 1 \), you calculate \( y = 3 + 2(1) = 5 \), giving you the point (1, 5).
• For \( x = -1 \), the calculation \( y = 3 + 2(-1) = 1 \) results in the point (-1, 1).

By plotting several points, you ensure that your line is accurate and supports the equation's characteristics.
Once these points are marked, draw a straight line through them to represent the full span of the linear function. Remember that linear functions extend infinitely in both directions, so make sure your line does too.

Linear Equation

Linear equations are foundational in algebra and describe straight lines when graphed. They have constant slopes and intercepts, giving them unique properties different from other kinds of equations. The equation we are working with, \( f(x) = 3 + 2x \), is a prime example.
The defining aspects of a linear equation are:

• It graphs as a straight line.
• The equation involves only the first power of the variable \( x \).
• The slope remains constant across the line.

Linear equations are not just confined to the slope-intercept form, though. They can also be represented as standard form or point-slope form. However, the slope-intercept form is often the easiest for graphing, allowing quick identification of the line's slope and starting point. Embracing these features of linear equations enables you to tackle a wide range of mathematical problems with confidence.