Problem 58 - Algebra 1: Concepts and Skills

Question

Graph the function. (Lesson 4.8) $$ g(x)=3 x+1 $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = 3x + 1\) is a straight line passing through the points (0,1) and (1,4). It has a slope of 3 and a y-intercept of 1.

Step 1: Identify the slope and y-intercept.

The standard form of a linear equation is \(y = mx + b\), where m is the slope and b is the y-intercept. Here, the slope \(m=3\) and the y-intercept \(b=1\).

Step 2: Plot the y-intercept.

Start by plotting the y-intercept on the y axis. Here the y intercept is 1, so plot a point at (0,1).

Step 3: Use the slope to find another point.

The slope is the rise over the run. Here our slope is 3/1 which means we go up 3 units and to the right 1 unit. From your first point (0,1), move 3 up and 1 to the right to plot your second point at (1,4).

Step 4: Draw a line through the points.

Finally, draw a straight line passing through both points. This line represents the graph of the function \(g(x) = 3x + 1\).

Key Concepts

Linear Equations

Linear equations are one of the simplest forms of equations you'll encounter in mathematics. They create a straight line on a graph, which corresponds to the equation's name. A linear equation can be identified because it involves only constants and a single variable with an exponent of one, such as \(ax + b = 0\). No variables are raised to a power higher than one in these equations.
This means there are no squares, cubes, or other types of polynomials involved. In two variables, the standard form of a linear equation is \(y = mx + b\), where \(y\) and \(x\) are your variables. The terms \(m\) and \(b\) represent constants that will be defined in the next section.
Linear equations are abundant in real-world applications, from calculating speed, investment returns, or even simple budgeting. Understanding how to work with them forms a vital foundation for further math concepts.

Slope-Intercept Form

The slope-intercept form is a specific way to express a linear equation. It is written as \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept. This form is particularly useful because it quickly provides insight into the graph’s characteristics.

• The slope \(m\) tells us how steep the line is and the direction it goes.
• The y-intercept \(b\) tells us where the line crosses the y-axis.

Starting with the slope-intercept form makes it easy to graph the equation. You just need to plot the y-intercept, then use the slope to determine other points on the line. It's intuitive and helpful for visualizing equations quickly when given in this format.

Plotting Points

Plotting points is essential for graphing linear equations accurately. Begin by identifying specific points from your equation to plot on the grid. Typically, you start with the y-intercept, because it’s the easiest to identify and plot.
For example, for the equation \(g(x) = 3x + 1\), the y-intercept \((0,1)\) gives you the first point. Marking points on the graph allows you to create an accurate representation of the equation's line. Each point has a specific x and y coordinate determining its position. After plotting the y-intercept, use the slope to find additional points.
This way, you ensure your line is correctly placed. Always remember: a good graph begins with correctly plotted points!

Slope of a Line

The slope of a line measures how a line rises or falls as it moves across the graph. When working with the linear equation \(y = mx + b\), the coefficient \(m\) indicates the slope. It expresses the rate of change, showing how much \(y\) changes for a change in \(x\).
In the equation \(g(x) = 3x + 1\), the slope \(m\) is 3. This means for every 1 unit you move to the right along the x-axis, the line moves up 3 units. The slope can also be thought of in terms of "rise over run"—how much you "rise" vertically versus how much you "run" horizontally.
Slopes can be positive, negative, zero, or undefined:

• Positive slope: line rises to the right.
• Negative slope: line falls to the right.
• Zero slope: line is horizontal.
• Undefined slope: line is vertical.

Understanding the slope is crucial as it impacts the direction and steepness of your line on the graph.