Problem 28 - Algebra 1

Question

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

Step-by-Step Solution

Answer

This linear graph will start at the y-intercept (0,4) and rise at a slope of 4. The line will cross through points like (0,4) and (1,8).

Step 1: Identify the slope and y-intercept

The equation provided is in the form of \(y=mx +b\). In this case, 'm', the slope, is 4 and 'b', the y-intercept is 4.

Step 2: Plot the y-intercept

The y-intercept is the point where the graph crosses the y-axis. Since 'b' is 4, go to 4 on the y-axis and place a dot. This is the first point on your graph.

Step 3: Use the slope to find another point

The slope is the rise over the run, which means we go up 4 units and move to the right by 1 unit from our y-intercept. Place a dot at this new location.

Step 4: Draw the line

Finally, draw a straight line connecting the two points. Your line should be going upwards, because you have a positive slope.

Key Concepts

Slope-Intercept Form

The equation of a line in slope-intercept form is one of the easiest ways to graph linear equations. This form is expressed as \( y = mx + b \). Let's break it down:

• \( y \) is the dependent variable, which you solve for in the equation.
• \( m \) is the slope, telling you how steep the line is.
• \( x \) is the independent variable.
• \( b \) is the y-intercept, where your line crosses the y-axis.

Many students find that this form makes it super simple to graph a line because it gives immediate visibility into the two key components—the slope (\( m \)) and the y-intercept (\( b \)). Once you identify these, plotting the graph becomes straightforward. Whether you're solving homework or preparing for a test, always try to get your linear equation into slope-intercept form first for an easier graphing experience.

Slope Calculation

When graphing a linear equation in slope-intercept form, calculating the slope is essential. The slope, represented by 'm' in the equation \( y = mx + b \), indicates the rise over the run between any two points on the line.

• Rise: This tells you how much you move up or down on the y-axis.
• Run: This refers to how far you move left or right on the x-axis.

For example, if your slope \( m \) is 4, this means you move up 4 units on the y-axis for every 1 unit you move to the right on the x-axis.
To find the slope without a graph, you can always rearrange a standard form equation into slope-intercept form. The slope is a critical component as it also tells you about the nature of the line:

• A positive slope means the line rises to the right.
• A negative slope indicates it falls to the right.
• A zero slope represents a horizontal line.
• An undefined slope signifies a vertical line.

Understanding slope can help you predict how the graph of your equation will look even before you start plotting points.

Y-Intercept Identification

Identifying the y-intercept is the crucial first step in plotting a linear equation graph. The y-intercept is where your line crosses the y-axis, and in the equation \( y = mx + b \), 'b' represents this point.
Finding this point is simple: Look at the equation and recognize the constant number that does not accompany an 'x'. For example, in \( y = 4x + 4 \), the y-intercept is 4.
This means your line will cross the y-axis at the point (0, 4).
When you begin graphing:

• Start by plotting your y-intercept on the y-axis.
• This serves as your anchor point for the rest of the graph.
• From there, use your slope to determine other points.

This initial step of identifying and plotting the y-intercept sets the foundation for accurately drawing the line. By making the y-intercept your first plotted point, the graphing process becomes more organized and methodical.