Problem 14 - Algebra Form and Function

Question

In Exercises \(14-17\) (a) Is \(y\) proportional, or is it inversely proportional, to a positive power of \(x\) ? (b) Make a table of values showing corresponding values for \(y\) when \(x\) is \(1,10,100,\) and 1000 . (c) Use your table to determine whether \(y\) increases or decreases as \(x\) gets larger. \(y=2 x^{2}\)

Step-by-Step Solution

Answer

Answer: The value of \(y\) increases as the value of \(x\) gets larger.

Step 1: (1) Determine proportionality or inverse proportionality

(To find out if y is proportional or inversely proportional to a positive power of x, we can analyze the given equation \(y = 2x^2\). Since the power of \(x\) is positive and y is directly connected to \(x^2\), we can conclude that \(y\) is proportional to a positive power of \(x\).)

Step 2: (2) Create a table of values for \(y\) when \(x = 1, 10, 100,\) and 1000

(Using the given equation, we can calculate the value of y for each given x value. Fill in the table with corresponding values of y: |x|y| |--|-| |1 |2(1)^{2} = 2 | |10|2(10)^{2} = 200 | |100|2(100)^{2} = 20,000 | |1000|2(1000)^{2} = 2,000,000 | Note that the results in the table are from plugging the x values into the expression \(y = 2x^2\).)

Step 3: (3) Determine whether y increases or decreases as x gets larger

(By examining the table, we can see that as x gets larger, so does the value of \(y\). Therefore, y increases as x gets larger.)

Key Concepts

Inverse Proportionality

Inverse proportionality is a fascinating concept in mathematics. It occurs when one quantity increases while the other decreases, maintaining a constant product. In other words, if two variables are inversely proportional, the product of these two variables is always the same regardless of any changes in their individual values.
For instance, consider the relationship between speed and travel time for a fixed distance. As speed increases, travel time decreases proportionally so that the product (speed × travel time) remains constant.

• Variables are connected by a constant product: If you multiply one variable by a certain factor, the other variable is multiplied by the reciprocal of that factor.
• This relationship can be expressed mathematically as: \( y \propto \frac{1}{x} \) or, equivalently, \( y \times x = k \), where \(k\) is a constant.
• Inverse proportionality is often seen in physics and engineering, especially in concepts involving rates or speeds.

It might initially seem complex, but the principle of opposites moving together helps clarify it. In our original exercise, the concept of inverse proportionality is not applied, as the variables are directly proportional.

Positive Power

Understanding positive power is essential when dealing with algebraic equations involving variables raised to certain powers. A positive power, often referred to as a positive exponent, shows how many times a number is multiplied by itself.
So, let's say you have an equation like \( y = x^n \), where \( n \) is a positive exponent:

• This means \( x \) is raised to the power of \( n \), which translates to multiplying \( x \) by itself \( n \) times.
• In our exercise, \( y = 2x^2 \), the variable x is raised to the second power, indicating a quadratic relationship.
• Positive powers indicate direct proportionality, as increasing the value of \( x \) results in a much larger increase in \( y\) depending on its power.

The most common positive power is the square, seen in many geometrical and physical world relationships. Understanding how powers work helps predict how changes in one variable affect another.

Algebraic Equations

Algebraic equations form the backbone of mathematics, enabling the representation of relationships between different quantities. These equations consist of variables, constants, and operations leading to expressions set equal to something.
The general form can look like this: \( ax^n + bx^{n-1} +...+ k = 0 \). In our exercise, we dealt with a simpler equation \( y = 2x^2 \), which can be broken down as follows:

• Variables (like \( x \)) represent unknown quantities that can vary.
• Coefficients (like \( 2 \)) are constants that multiply the variables.
• Exponents (like the \( 2 \) in \( x^2 \)) are powers indicating how many times a variable is multiplied by itself.
• Even the simplest algebraic equations can provide deep insights into relationships and changes.

By solving algebraic equations, you can find unknown values and understand how changing one element influences another. This understanding is integral to fields like physics, engineering, and economics, where such equations model real-world scenarios efficiently.