Problem 17 - Algebra Form and Function

Question

(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. \(y=\frac{5}{x^{2}}\)

Step-by-Step Solution

Answer

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

Step 1: Determine the type of proportionality

The given function is \(y=\frac{5}{x^{2}}\). Since \(y\) is in the form of a quotient with \(x\) in the denominator, it shows that \(y\) is inversely proportional to a positive power of \(x\) (in this case, power 2).

Step 2: Creating a table of values for \(y\)

Plug in the values \(x = 1, 10, 100,\) and 1000 into the function \(y=\frac{5}{x^{2}}\) to find the corresponding \(y\) values: For \(x=1\): \(y = \frac{5}{1^2} = 5\) For \(x=10\): \(y = \frac{5}{10^2} = \frac{5}{100} = 0.05\) For \(x=100\): \(y = \frac{5}{100^2} = \frac{5}{10000} = 0.0005\) For \(x=1000\): \(y = \frac{5}{1000^2} = \frac{5}{1000000} = 0.000005\) The table of values is: | x | y | |-------|-----------| | 1 | 5 | | 10 | 0.05 | | 100 | 0.0005 | | 1000 | 0.000005 |

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

From the table, we can see that as \(x\) increases, the value of \(y\) decreases. Hence, \(y\) decreases as \(x\) gets larger.

Key Concepts

Proportionality

In mathematics, proportionality describes a relationship where one quantity is related to another by a constant. Consider two types of proportionality: direct and inverse.

• Direct Proportionality: If two variables are directly proportional, as one increases, so does the other. This relationship can be expressed by the formula \( y = kx \), where \( k \) is a constant.
• Inverse Proportionality: Alternatively, when two variables are inversely proportional, as one increases, the other decreases. The mathematical expression for this is \( y = \frac{k}{x^n} \), with \( n \) usually denoting the power of \( x \).

In the problem (a) from the exercise, we see \( y = \frac{5}{x^2} \). Here, \( y \) is inversely proportional to \( x^2 \), meaning as \( x \) increases, \( y \) decreases. Understanding this is crucial for evaluating the behavior of functions like these.

Exponentiation

Exponentiation involves raising a number to a power, represented as \( x^n \), where \( x \) is the base and \( n \) is the exponent. The exponent indicates how many times the base multiplies itself. For example, for \( x^2 \), \( x \) is multiplied by itself once, so \( x^2 = x \times x \).This concept is pivotal to the exercise since the function is \( y = \frac{5}{x^2} \). The denominator, \( x^2 \), means \( x \) is squared. As demonstrated in the exercise, substituting various values of \( x \) alters the result because the square grows rapidly, drastically influencing \( y \). Each increase in \( x \) significantly decreases \( y \), emphasizing the impact of exponentiation on inverse proportionality relationships.

Function Tables

Function tables help organize data corresponding to values of \( x \) and \( y \) in relation to the given function, making it easier to discern patterns and relationships.In the exercise, we used a function table to plug in values of \( x = 1, 10, 100, \) and \( 1000 \) into \( y = \frac{5}{x^2} \). Here's how the values came out:

• For \( x = 1 \), \( y = 5 \)
• For \( x = 10 \), \( y = 0.05 \)
• For \( x = 100 \), \( y = 0.0005 \)
• For \( x = 1000 \), \( y = 0.000005 \)

These values clearly show the inverse relationship: as \( x \) grows, \( y \) shrinks. Using function tables makes tracking these changes straightforward and helps ascertain whether the function displays a trend of increase or decrease for \( y \) as \( x \) changes.