Problem 39 - Precalculus Essentials

Question

Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$y=2 x+1$$

Step-by-Step Solution

Answer

The slope of the line is 2 and the y-intercept is 1. The line passes through (0,1) and rises 2 units for each 1 unit it moves to the right.

Step 1: Identify the Slope

In the given equation \(y = 2x + 1\), the coefficient of \(x\) is 2, thus the slope \(m\) is 2.

Step 2: Identify the Y-Intercept

The constant term in the equation is \(1\), thus the \(y\)-intercept \(b\) is 1.

Step 3: Graph the Function

Begin graphing the function by plotting the \(y\)-intercept, (0, 1). From this point, use the slope to find other points on the line. The slope of 2 means to go up 2 units and over 1 unit to the right. Continue this pattern to identify additional points. Draw a line through the identified points to represent the function \(y = 2x +1\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when studying linear functions. The slope is a measure of how steep the line is. Mathematically, it's defined as the rise over the run, or the change in the 'y' value divided by the change in the 'x' value between two distinct points on the line.

In the equation \(y = 2x + 1\), the slope is represented by the coefficient of \(x\), which is 2. This number tells us for each step we move to the right along the x-axis, the value of \(y\) increases by 2 units. If the slope was negative, the line would go down as we move from left to right. The concept of slope is foundational in understanding how to analyze and graph linear relationships.

Y-intercept

The y-intercept of a graph is the point at which the line crosses the y-axis. This is where the value of \(x\) is zero. For the equation \(y = 2x + 1\), the y-intercept is identified in the constant term, which is 1.

Recognizing the y-intercept allows us to start plotting the graph at the point (0, 1) and is significant because it provides a starting place for drawing our line. Every linear function in the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept, can be easily graphed using these two critical pieces of information.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when plotted on a graph. They have one or more variables, but each term is either a constant or the product of a constant and a single variable.

The general form of a linear equation is \(y = mx + b\), where \(m\) denotes the slope and \(b\) denotes the y-intercept. In our sample problem, the linear equation \(y = 2x + 1\) follows this format with a clear slope and y-intercept. Linear equations are essential for modeling real-world scenarios with a constant rate of change, among numerous other applications in various fields.

Plotting Linear Graphs

Plotting linear graphs is a straightforward process once you know the slope and y-intercept. Begin at the y-intercept, which is our starting point on the graph. For the example \(y = 2x + 1\), you would start at the point (0,1).

Next, use the slope to determine the direction and steepness of your line. With a slope of 2, you move up 2 units in the y direction for every 1 unit you move to the right in the x direction. The 'up 2, right 1' movement allows you to plot additional points. Once you have two or more points, draw a straight line through them to complete the graph. This visual depiction of the linear equation helps to understand the concept of slope and y-intercept practically.