Problem 15 - Algebra Form and Function

Question

Rewrite the function in slope-intercept form. $$ f(x)=12+3(x-1) $$

Step-by-Step Solution

Answer

Question: Rewrite the function $$f(x) = 12 + 3(x - 1)$$ in slope-intercept form and identify the slope and y-intercept. Answer: The function in slope-intercept form is $$f(x) = 3x + 9$$. The slope (m) is 3, and the y-intercept (b) is 9.

Step 1: Distribute the constant to the expression inside the parentheses

In order to rewrite the function in slope-intercept form, we first need to distribute the constant (3) to the expression inside the parentheses \((x - 1)\). $$ f(x) = 12 + 3(x - 1) $$ Let's distribute the constant: $$ f(x) = 12 + 3x - 3 $$

Step 2: Combine like terms

Now, we can combine the constant terms (12 and -3) in the equation: $$ f(x) = 3x + 9 $$

Step 3: Rewrite the function in slope-intercept form

Since we have simplified the function into the form $$f(x) = 3x + 9$$, we can identify the slope and y-intercept directly. The equation is already in slope-intercept form, with $$m = 3$$ (the slope) and $$b = 9$$ (the y-intercept). The final answer is: $$ f(x) = 3x + 9 $$ This is the slope-intercept form of the given function.

Key Concepts

Understanding Linear Equations

Linear equations are equations of the first order. They describe a straight line when graphed on a coordinate plane. Linear equations can be expressed in the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis.

These equations are very straightforward and are often used in various fields such as physics, economics, and statistics because they provide a simple way to model relationships between two variables. In mathematics, linear equations are fundamental, and understanding them helps in analyzing and plotting data efficiently.

In our scenario, we begin with an equation that doesn't initially appear in textbook linear form. To convert it to slope-intercept form, we need to manipulate the terms using algebraic principles.

The Art of Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to uncover desired structures or solve for a variable. In our provided exercise, the function \(f(x)=12+3(x-1)\) is our starting point. This expression isn't initially in slope-intercept form, so an algebraic process is necessary.

Here's a simple guide to perform this process using our example:

• Step 1: Distribute multiplication over subtraction. Use the distributive property to multiply \(3\) by each term inside the parentheses \((x - 1)\). This results in \(3x - 3\). The function now reads \(f(x) = 12 + 3x - 3\).
• Step 2: Combine like terms. With \(12 + 3x - 3\), identify and merge the constant terms. Adding \(12\) and \(-3\) gives \(9\), so the function looks like: \(f(x) = 3x + 9\).

By mastering these steps, breaking down an equation into a more understandable form becomes second nature, paving the way to easier problem-solving.

Exploring Function Transformation

Function transformation is the process of altering the standard form of a function to understand its behavior more deeply. When a function is expressed in different forms, the focus often shifts to different attributes, helping to visualize operations geometrically.

In our example, we transform the original function \(f(x)=12+3(x-1)\) into the slope-intercept form \(f(x) = 3x + 9\). This transformation emphasizes how the function behaves visually once plotted:

• Slope \((m)\): The slope value \(3\) tells how steeply the line ascends or descends. A positive slope like ours indicates a line that rises to the right.
• Y-intercept \((b)\): The y-intercept \(9\) signifies the point at which the line will cross the y-axis, rendering it essential for graphing purposes.

Understanding these transformations allows one to depict real-world situations with linear models more effectively, making concepts tangible and comprehensible.