Problem 41 - Algebra Form and Function

Question

Give the slope and \(y\) -intercept for the graphs of the functions. $$ f(x)=\frac{20-2 x}{3} $$

Step-by-Step Solution

Answer

Answer: The slope of the function is \(-\frac{2}{3}\) and the y-intercept is \(\frac{20}{3}\).

Step 1: Rewrite the function in slope-intercept form

To rewrite the function in slope-intercept form, we need to show it as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. The given function is: $$ f(x)=\frac{20-2x}{3} $$ Since \(f(x)\) represents the y-value, we can rewrite the function as: $$ y=\frac{20-2x}{3} $$

Step 2: Distribute the denominator

Now we'll distribute the denominator (3) to simplify the equation: $$ y = \frac{20}{3} - \frac{2x}{3} $$

Step 3: Identify the slope and y-intercept

The equation is now in slope-intercept form, which is \(y = mx + b\). The slope \(m\) is the coefficient of the \(x\) term, and the y-intercept \(b\) is the constant term. In this case, the slope and y-intercept are: $$ m = -\frac{2}{3}, \quad b = \frac{20}{3} $$ The slope of the function is \(-\frac{2}{3}\) and the y-intercept is \(\frac{20}{3}\).

Key Concepts

Understanding the Slope

In the world of linear equations, the slope is a fundamental concept that tells us about the steepness of a line. The slope is represented by the letter \(m\) in the slope-intercept form equation \(y = mx + b\). It indicates how much \(y\) changes for a given change in \(x\).
For instance, if the slope is positive, the line rises as it moves from left to right. If the slope is negative, it falls. In our example, with \(f(x) = \frac{20-2x}{3}\), we found that the slope \(m\) is \(-\frac{2}{3}\). This means that for every increase of 3 units in \(x\), \(y\) decreases by 2 units.
A steeper slope (larger absolute value) indicates a line tilts more sharply, while a smaller slope means it's more gradual. Understanding the slope helps us predict how the line behaves in relation to the coordinate axes. Always remember, slope shows rate of change!

Decoding the Y-Intercept

The y-intercept is another key concept in linear equations, which indicates where the graph crosses the y-axis. In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept. It occurs when \(x = 0\).
In our equation \(f(x) = \frac{20-2x}{3}\), after rearranging, we identify \(b\) as \(\frac{20}{3}\). This tells us that the line will intersect the y-axis at the point \((0, \frac{20}{3})\).
The y-intercept helps us understand where a line starts on the graph. From this point, the slope guides the rest of the line's path. Knowing the y-intercept is especially handy when graphing a line, as it provides a concrete starting point on the y-axis.

Grasping Linear Equations

Linear equations are equations of a straight line and can be expressed in the form \(y = mx + b\). They are called 'linear' because when graphed, they produce a straight line.
The equation consists of two primary components: the slope \(m\), indicating the line's steepness, and the y-intercept \(b\), showing the starting point on the y-axis.

• Slope \(m\): rate of change of the function.
• Y-intercept \(b\): the value of \(y\) when \(x = 0\).

Linear equations are fundamental in algebra and appear in many real-world applications such as calculating expenses or projecting growth.
For example, with our original function \(f(x) = \frac{20-2x}{3}\), by deriving its slope \(-\frac{2}{3}\) and y-intercept \(\frac{20}{3}\), we form a complete picture of the line's behavior. With these values, you can easily graph it or apply the equation to various scenarios. Understanding this form enhances your ability to interpret and create linear models.