Problem 45 - Algebra A Combined Function

Question

Graph each equation. See Sections 3.2 and 3.3. $$ y=2 x+5 $$

Step-by-Step Solution

Answer

Graph the line with a slope of 2 and a y-intercept of 5, crossing at points (0, 5) and (1, 7).

Step 1: Identify the slope and y-intercept

The given equation is in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope \(m = 2\) and the y-intercept \(b = 5\).

Step 2: Plot the y-intercept

Start graphing by plotting the y-intercept \((0, 5)\) on the y-axis. This point is where the line crosses the y-axis.

Step 3: Use the slope to find another point

The slope \(m = 2\) means that for every step of 1 unit to the right on the x-axis, the line rises by 2 units. From the y-intercept \((0, 5)\), move 1 unit to the right and 2 units up to find a new point, \((1, 7)\). Plot this point on the graph.

Step 4: Draw the line

Use a ruler to draw a straight line through the points \((0, 5)\) and \((1, 7)\). Extend the line in both directions, adding arrows to indicate that the line goes on infinitely.

Key Concepts

slope-intercept form

The slope-intercept form is a standard way of expressing a linear equation in mathematics. It is given by the formula \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) stands for the y-intercept, which is where the line crosses the y-axis.
Writing an equation in this form can make it easier to graph, as you can instantly identify the slope and y-intercept. It allows for quick sketching of the graph by easily plotting the y-intercept and using the slope to find other points.
The brilliant thing about the slope-intercept form is that it provides a clear picture of how the line behaves across the graph. With \(y = 2x + 5\), for example, you can see that the slope is 2, and the line intersects the y-axis at 5. This insight is valuable when analyzing the relationships between variables in the equation and visualizing their graphical representation.

slope

Slope is a measure of the steepness of a line. In the slope-intercept form \(y = mx + b\), the slope is indicated by \(m\). It tells you how much the line rises or falls as you move to the right along the x-axis.
Remember that the slope is a ratio of the vertical change (rise) over the horizontal change (run) between two points on the line. It is expressed as \(\frac{\text{rise}}{\text{run}}\).
For our equation \(y=2x+5\), the slope is 2. This means for every move 1 unit right on the x-axis, the line moves 2 units up. This steady rise gives the line a positive incline, signaling that as x increases, the y-value also increases. A steeper slope indicates a greater rate of change, which is very evident in graphical interpretations.

y-intercept

The y-intercept is a critical component in interpreting the position of a line on a graph. In the equation \(y = mx + b\), \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. This occurs when \(x = 0\).
In our example equation \(y = 2x + 5\), the y-intercept is 5. This tells us that the line will pass through the point \((0, 5)\) on the y-axis.
Plotting the y-intercept is the first step in graphing because it gives us a starting point. From here, you can use the slope to find additional points on the graph, helping to lay the foundation for drawing the entire line. Understanding this concept is key for graphing linear equations efficiently.