Problem 42 - Algebra Form and Function

Question

Give the slope and \(y\) -intercept for the graphs of the functions. $$ f(x)=15-2(3-2 x) $$

Step-by-Step Solution

Answer

Answer: The slope (m) of the linear function is 4 and the y-intercept (b) is 9.

Step 1: Simplify the function

Start by simplifying the function using the distributive property: $$ f(x) = 15 - 2(3 - 2x) = 15 - 6 + 4x $$ Now, combine like terms: $$ f(x) = 9 + 4x $$

Step 2: Identify the slope and y-intercept

Now that the function is in the form \(f(x) = 4x + 9\), it's easy to see that the slope (\(m\)) is 4 and the y-intercept (\(b\)) is 9. To summarize: Slope (\(m\)): 4 Y-intercept (\(b\)): 9

Key Concepts

Slope

When you look at a linear function like \( f(x) = mx + b \), the slope is a key feature. It tells you about the steepness and direction of the line. The slope is represented by \( m \). A larger positive value for \( m \) means the line rises steeply as you move from left to right. Conversely, if \( m \) is negative, the line falls as it moves left to right.
The slope can be thought of as "rise over run," which means how much the line goes up or down over a certain horizontal distance. With a function rewritten as \( f(x) = 4x + 9 \), the slope is 4.

• If the slope is 0, the line is perfectly horizontal.
• If the slope is undefined, the line is vertical.

Understanding the slope helps predict how one variable affects another in real-world scenarios.

Y-intercept

The y-intercept is where the line crosses the y-axis in a graph. In the function \( f(x) = mx + b \), \( b \) is the y-intercept. It tells you the value of \( f(x) \) when \( x = 0 \). It's the point where the line meets the vertical axis.
In the example function \( f(x) = 4x + 9 \), the y-intercept is 9. This means that when \( x = 0 \), \( f(x) = 9 \). The line will intersect the y-axis at the point (0, 9).

• The y-intercept gives you a starting point on the graph.
• In the context of word problems, it often represents a fixed value or starting quantity.

Remember, it's crucial for graphing linear functions effectively and determining initial conditions in applications.

Distributive Property

The distributive property is a useful mathematical rule for simplifying expressions. It lets you multiply a single term by each term inside a parenthesis. Essentially, you are distributing the multiplication over addition or subtraction inside the parenthesis.
For example, in the function \( f(x) = 15 - 2(3 - 2x) \), the distributive property helps expand the equation. Here's how it works: you multiply \(-2\) by both \(3\) and \(-2x\).

• First, \(-2 \times 3 = -6\).
• Next, \(-2 \times -2x = 4x\).

This simplifies the expression to \( f(x) = 15 - 6 + 4x \).
Using the distributive property correctly helps to break down and simplify algebraic expressions, making them easier to handle in subsequent steps.