Problem 30 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=23 $$

Step-by-Step Solution

Answer

The slope of the graph of \(f(x) = 23\) is 0. This indicates the function is constant.

Step 1: Understand the Function

The function given is a constant function, which means it does not change with different values of \(x\). It is given by \(f(x) = 23\). This means that for any value of \(x\), \(f(x)\) will always be 23.

Step 2: Identify the Graph Type

A constant function like \(f(x) = 23\) is represented by a horizontal line in the graph. It is parallel to the x-axis and passes through all points where the y-coordinate is 23.

Step 3: Determine the Slope

The slope of a line is calculated as the change in the y-coordinate divided by the change in the x-coordinate, essentially \(\frac{\Delta y}{\Delta x}\). For a horizontal line, there is no change in the y-coordinate as x changes. Thus, the slope of the graph is 0.

Step 4: Interpret the Slope

A slope of 0 means that the function does not increase or decrease as x increases. It represents a horizontal line, indicating that the output of the function remains constant regardless of changes in the input.

Key Concepts

Constant Function

A constant function is a special type of function in mathematics. In simple terms, it means that no matter what value we give to the variable, the function always returns the same result. For example, if we have a function like \( f(x) = 23 \), this shows us that for any input \( x \), the output, or \( f(x) \), is always 23. Unlike other functions that change and can represent motion or growth, constant functions are stable and unchanging.

Some important features of constant functions are:

• They have no dependence on the variable's value. This makes calculations and graphs relatively simple.
• Their expressions never include the variable in a way that alters the result. In other words, the expression remains fixed.

Constant functions often appear in real-world scenarios where certain measures need to remain stable or unchanged, like a fixed interest rate.

Horizontal Line

When we talk about the graph of a constant function like \( f(x) = 23 \), we can visualize it as a horizontal line. This means it stretches left to right across the graph without ascending or descending.

Here’s what we mean when we say a line is horizontal:

• It runs parallel to the \( x \)-axis.
• Every point on the line has the same \( y \)-coordinate, which in this example would be 23.
• The line does not tilt upward or downward, staying completely flat.

Horizontal lines are unique in how they appear on graphs:

• Their consistency is a direct result of the constant value of the function.
• They demonstrate situations where outcomes do not fluctuate with input changes.

Recognizing a horizontal line on a graph is crucial because it tells you that the function has a slope of zero, which is another core concept we'll explore.

Interpreting Slope

The slope of a line provides us with important information about a function. It essentially tells us how steep a line is, or in simpler terms, how much it goes up or down as it moves from left to right.

For a constant function like \( f(x) = 23 \), the slope is calculated as zero. Here’s why:

• Slope is typically defined as the ratio \( \frac{\Delta y}{\Delta x} \), which measures vertical change over horizontal change.
• In this case, since \( y \) (or \( f(x) \)) is always the same, \( \Delta y = 0 \) regardless of how \( \Delta x \) changes.

Understanding a slope of zero is key because it tells us:

• The function is constant, meaning no matter how much we increase or decrease our \( x \) values, \( f(x) \) remains constant.
• Such a slope confirms our graph is a horizontal line, indicating no movement up or down.

This interpretation is crucial in fields like economics and physics, where realizing that a factor remains unchanged helps simplify analysis and decision-making.