Problem 4 - Applied Calculus

Question

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\frac{3}{8 x}$$

Step-by-Step Solution

Answer

The function is a power function with \( k = \frac{3}{8} \) and \( p = -1 \).

Step 1: Review Function Format

The given function is \( y = \frac{3}{8x} \). To determine if a function is a power function, the function should be expressed in the form \( y = kx^p \), where \( k \) is a constant and \( p \) is a real number exponent.

Step 2: Rewrite the Function

Rearrange the given function into the form \( y = kx^p \). Start by rewriting the denominator \( 8x \) as a single term with a negative exponent: \( 8x = 8 \times x^1 \). So, \( \frac{1}{8x} = \frac{1}{8} \times x^{-1} \). Then, the function becomes: \( y = \frac{3}{8} \times x^{-1} \).

Step 3: Identify Constants and Exponents

Comparing \( y = \frac{3}{8} x^{-1} \) with \( y = kx^p \), we identify \( k = \frac{3}{8} \) and \( p = -1 \).

Step 4: Determine if Power Function

Since the function can be expressed in the form \( y = kx^p \) with real constants \( k \) and \( p \), it is a power function. In this case, \( y = \frac{3}{8} x^{-1} \).

Key Concepts

Constant

In mathematics, a constant is a fixed value that does not change. In the context of power functions, a constant refers to the coefficient placed in front of the variable, such as the "3/8" in our given function, \( y = \frac{3}{8} x^{-1} \). 💡Here's a friendly tip: the constant \( k \) provides a multiplication factor that scales the entire function.

When we express power functions in the form \( y = kx^p \), the constant \( k \) helps us understand how steep or flat the graph of the function might be.

Why is understanding constants important?

• They determine the overall shape and direction of our function's graph.
• They allow us to compare different functions in an intuitive manner.

Remember, if you see a function with constants, you know that part of that function is steadfast and unchanging, like a solid rock ensuring stability within the equation.

Exponent

An exponent, put simply, is the power to which a number, known as the base, is raised. In our case, the exponent in the function \( y = \frac{3}{8} x^{-1} \) is \(-1\). This indicates the operation we perform on our base (in this scenario, the variable \( x \)). Wondering why exponents matter? 🔍 Here's why:

Exponents are crucial because they indicate repeated multiplication. When you see an exponent, think of it as giving you a shortcut to multiply a number by itself "p" times. But sometimes, like in our exercise, the exponent can be negative. More on that in a bit.

Also, understanding how exponents work can help you:

• Simplify complex mathematical expressions.
• Quickly estimate large numbers and operations.
• Unlock the mystery behind many real-world phenomena described by exponential growth or decay.

The exponent is like the key that turns a simple variable into something much more exciting and complex!

Negative Exponent

Negative exponents can seem mysterious, but they're not as intimidating once you get the hang of them. In the function \( y = \frac{3}{8} x^{-1} \), the exponent of \( -1 \) implies a specific mathematical operation.

So what does a negative exponent mean? 📉 The rule to remember is:

• \( x^{-n} = \frac{1}{x^n} \)

This simply means the negative exponent tells you to take the reciprocal of the base.
In our case, \( x^{-1} \) is equivalent to \( \frac{1}{x^1} \), transforming our function into a more approachable form.

Understanding negative exponents is handy because:

• They help simplify fractional or complex equations by turning problems with division into equivalent problems with multiplication.
• They often appear in formulas modelling decay processes in real life, like depreciation over time.

Negative exponents open the door to seeing numbers from a different perspective, shifting intuition into deeper mathematical insight.

Function Transformation

Function transformation involves shifting, stretching, or flipping the graphs of functions. When analysing, or altering functions, spotting transformations can make complex mathematics simpler.

Looking at our example, \( y = \frac{3}{8} x^{-1} \), we see a transformation via a negative exponent. This changes the simple graph of \( x^1 \) into something more intriguing.

Transformations can include:

• Vertical or horizontal shifts, moving the entire graph up/down or left/right.
• Scaling, which stretches or compresses the graph vertically or horizontally.
• Reflection, such as flipping the graph over an axis.

In the classroom of function transformation 📚, you're able to:

• Recognise the effects specific operations have on the graph of a function.
• Predict the graphical outcome of function alterations.

Function transformations are like math's creative freedom, allowing a single equation to take multiple forms with a few tweaks!