Problem 16 - Algebra Form and Function

Question

Rewrite the function in slope-intercept form. $$ f(x)=1800+500(x+3) $$

Step-by-Step Solution

Answer

Answer: The slope (m) is 500 and the y-intercept (b) is 3300.

Step 1: Distribute the expression

First, distribute the 500 to each term inside the parentheses: $$ f(x) = 1800 + 500(x + 3) = 1800 + 500x + 1500 $$

Step 2: Combine like terms

Now, combine the constants to simplify the equation: $$ f(x) = 500x + (1800 + 1500) = 500x + 3300 $$

Step 3: Put in slope-intercept form

The function is now in slope-intercept form: $$ y = 500x + 3300 $$ Here, the slope (m) is 500, and the y-intercept (b) is 3300.

Key Concepts

Distributive Property

The distributive property is a crucial mathematical concept often used while dealing with expressions involving parentheses. It's like handing out tasks to all members of a group. Imagine you have a number outside the parentheses, as in the expression \(500(x+3)\). The distributive property allows you to distribute, or "hand out," this 500 to each term inside the parentheses.

Here's what it means in practice:

• You multiply 500 by \(x\), resulting in \(500x\)
• Then, you multiply 500 by 3, resulting in 1500

By using the distributive property, our expression \(500(x+3)\) becomes \(500x + 1500\).

This step simplifies the expression and sets you up for further simplification, like combining like terms.

Combining Like Terms

Combining like terms is an important step in simplifying an expression. It helps to tidy things up, making the equation more straightforward to read and work with. In algebra, like terms have the same variable raised to the same power. For example, in the expression we discussed, \(1800 + 500x + 1500\), the terms 1800 and 1500 are both constants.

Here's how you combine like terms:

• Add the constant 1800 to 1500
• This results in a new term, 3300

Combining these terms, the expression now reads as \(500x + 3300\). This simplifies the original expression and prepares it for the final step of rewriting it in slope-intercept form.

Linear Function

A linear function is a mathematical function that graphs to a straight line. One of the most common ways to express a linear function is in slope-intercept form, represented as \(y = mx + b\). Here, \(m\) stands for the slope, showing how steep the line is, and \(b\) represents the y-intercept, indicating where the line crosses the y-axis.

In our example, we've transformed the given function into \(y = 500x + 3300\). This expression satisfies the slope-intercept form:

• "y" indicates the output of the function
• "500" is the slope, meaning for every 1 unit increase in \(x\), \(y\) increases by 500
• "3300" is the y-intercept, where the line hits the y-axis at 3300

This transformation allows us to easily understand and graph the behavior of the function by providing a clear visual representation on the coordinate plane.