Problem 39 - College Algebra 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, the \(y\)-intercept of the line is 1. The line can be graphed by starting at the \(y\)-intercept, (0,1) and plotting subsequent points using the slope, such as (1,3)

Step 1: Identify the Slope

Observe the given equation \(y = 2x + 1\). In comparison to the general form, \(y = mx + c\), we can see that the slope \(m\) is 2.

Step 2: Identify the \(y\)-Intercept

Similarly in the equation, the \(y\) -intercept \(c\) is 1.

Step 3: Plot the Function

To graph the linear function, start by marking the \(y\) -intercept, which is at point (0,1) on the \(y\)-axis. Then, use the slope to plot another point. The slope 2 means for each 1 unit increase in \(x\), \(y\) increases by 2 units. So, from (0,1), move right 1 unit and up 2 units to reach the point (1,3). Draw a straight line through these 2 points to complete the graph.

Key Concepts

Slope-Intercept Form

Understanding how to work with linear equations can be significantly simplified by using the slope-intercept form, which is a way to write linear equations to make them easy to graph. The standard format is \( y = mx + b \), where \(m\) is the slope of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis.

In the given exercise, the equation \(y = 2x + 1\) is already in the slope-intercept form. Here, you can easily identify the slope and the y-intercept, enabling a straightforward way to graph the function. This form presents a direct relationship between the equation and the visual representation on a graph.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis of a coordinate plane. In the context of the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).

For the equation provided, \(y = 2x + 1\), the y-intercept is 1. This means that the point (0, 1) is where the line will intersect the y-axis. Identifying the y-intercept is crucial as it provides a starting point for graphing the linear equation and represents the value of \(y\) when \(x = 0\).

Slope of a Line

The slope of a line is a measure of how steep the line is and the direction it travels on the coordinate plane. Calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line, it is often expressed as \(m\) in the slope-intercept equation \(y = mx + b\).

Rise over Run

A slope of 2, as seen in the example \(y = 2x + 1\), signifies that for every unit increase in \(x\) (run), \(y\) increases by 2 units (rise). This tells us the line angles upward as it moves from left to right across the graph. A negative slope indicates that the line angles downward.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations produce straight lines when graphed, and they have one or two variables, usually \(x\) and \(y\).

They can take on various forms such as standard form (\(Ax + By = C\)) and point-slope form (\(y - y_1 = m(x - x_1)\)), but for graphing purposes, the slope-intercept form is typically the most straightforward.

Consistency of Slope

Regardless of which two points on the line you choose to calculate it, the slope of a linear equation remains constant. This consistent rate of change is what keeps the graph of the equation straight and not curved.