Problem 6 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ y=11 x $$

Step-by-Step Solution

Answer

The slope of the graph of the function \(y = 11x\) is 11 and the y-intercept is 0.

Step 1: Finding the Slope

In this case, it can be noticed from the equation \(y = 11x\), that the slope is 11, because the slope \(m\) of a line in the slope-intercept form \(y = mx + b\) is the coefficient of \(x\). So, the slope \(m = 11\).

Step 2: Finding the Y-Intercept

The y-intercept is the value of \(y\) where the line intercepts the y-axis. This in the equation \(y = mx + b\) is represented by \(b\). Since there is no constant term in the equation, the y-intercept \(b = 0\).

Key Concepts

Slope of a Line

Understanding the slope of a line is essential for graphing linear equations and interpreting the rate at which one quantity changes with respect to another. The slope represents the steepness or incline of the line and is defined as the ratio of the rise (the vertical change) over the run (the horizontal change) between any two points on the line.

When we see a linear equation in the form of \(y = mx\), where \(m\) and \(x\) are variables, \(m\) represents the slope. In the exercise \(y = 11x\), the coefficient of \(x\) is 11, telling us that for each single unit increase in \(x\), the value of \(y\) increases by 11 units. Hence, the slope, or \(m\), is 11. This also means the line is quite steep, ascending rapidly as one moves from left to right along the graph.

Y-Intercept

The y-intercept of a graph is one of the key characteristics that can be quickly identified from an equation of a line. It is the point at which the line crosses the y-axis. This point indicates the value of \(y\) when \(x\) is 0.

In the equation form \(y = mx + b\), the \(b\) represents the y-intercept. If no \(b\) is explicitly presented, as in our exercise, it means that the line passes through the origin \((0,0)\) and thus, the y-intercept is 0. It is the starting point of the line on the y-axis, so in our example, the y-intercept tells us that when \(x = 0\), \(y\) will also be 0.

Slope-Intercept Form

The slope-intercept form of a equation of a line is one of the most direct ways to write out a linear equation. It's expressed as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. This form is very useful because it provides immediate information just from the coefficients.

For instance, in the example \(y = 11x\), this is actually in slope-intercept form with \(m = 11\) and \(b = 0\). This gives us instant insight into the properties of the line: how steep the line is, and where it crosses the y-axis.

Graphing Using Slope-Intercept Form

To graph a line using its slope-intercept form, you can start by plotting the y-intercept on the y-axis. Then, using the slope, you can determine the direction and steepness of the line by moving up or down (rise) and left or right (run) from the y-intercept point accordingly.