Problem 32 - Algebra 1: Concepts and Skills

Question

Graph the equation. $$y=4 x+4$$

Step-by-Step Solution

Answer

The graph of the equation \(y = 4x + 4\) is a straight line that passes through the points (0,4) and (1,8). The line slopes upward as it moves to the right, reflecting the positive slope.

Step 1: Analyze the Linear Equation

Understand that the linear equation provided is in slope-intercept form \(y=mx+b\). In this case, for the equation \(y=4x+4\), the slope \(m\) is 4 and the y-intercept \(b\) is also 4.

Step 2: Identify the y-intercept

Next, locate the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For this equation, the y-intercept is \(b=4\). This is the point (0, 4).

Step 3: Plot the y-intercept

Start by plotting the y-intercept on the graph. Mark a point at (0, 4).

Step 4: Use the slope to find another point

The slope is \(m=4\) which can be thought as 4/1. That means, for every step one unit to the right along the x-axis (positive direction), move four units up along the y-axis. Starting from the y-intercept (0, 4), move one unit to the right to x=1, and then move four units up. This gives another point on the line, (1, 8).

Step 5: Plot the second point and draw the line

Plot the point (1, 8) on the graph. Now that we have two points, we can draw a straight line through these points which represents the graph of the equation.

Key Concepts

Slope-intercept form

The slope-intercept form is a specific way to express the equation of a line. It is written as \(y = mx + b\).
This form is very useful for graphing because it directly tells you the slope and the y-intercept of the line.
In the equation \(y = 4x + 4\), we can easily identify:

• \(m\) as the slope, which is 4
• \(b\) as the y-intercept, which is also 4

The slope-intercept form is advantageous because it shows how the line behaves:
\(m\) (the slope) shows how steep the line is, and \(b\) indicates where the line crosses the y-axis.
This form makes it simpler to plot a line on a graph starting with the y-intercept.
By transforming any linear equation into this form, you can easily determine these two key characteristics of the line.

Y-intercept

The y-intercept is a critical part of plotting linear equations. It is the point where the line intersects the y-axis.
This concept is illustrated in the slope-intercept form of a line equation, \(y = mx + b\), by the \(b\) value.
For our example, the y-intercept of \(y = 4x + 4\) is 4. This means the line crosses the y-axis at the coordinate point (0, 4).

• The x-coordinate is 0 because it is the point on the y-axis.
• The y-coordinate is determined by the value of \(b\), which is 4 in this case.

To plot this on a graph, simply point to the coordinate (0, 4).
This point serves as a starting point from which you can use the slope to find additional points on the line.

Slope of a line

The slope of a line describes its steepness and direction. It is a ratio that shows how the line moves from one point to another.
This ratio is expressed in the slope-intercept form \(y = mx + b\) by the \(m\) value. For the equation \(y = 4x + 4\), the slope \(m\) is 4.
This value can be expressed as \(\frac{4}{1}\), translating into the line rising 4 units for every 1 unit it moves to the right.
This is often described as "rise over run."

• A positive slope (like 4) indicates the line travels upward as it moves from left to right.
• A negative slope would mean the line travels downward.

To graph this, start at the y-intercept (0, 4), then move one unit to the right along the x-axis and four units up along the y-axis to reach a new point.
By repeating this step, you establish the line's direction, making it easy to draw it on the graph.