Problem 27 - Algebra 1: Concepts and Skills

Question

DIRECT OR INVERSE VARIATION Make a table of values for \(x=-4,-3\) \(-2,-1,1,2,3,\) and \(4 .\) Use the table to sketch the graph. State whether \(x\) and \(y\) vary directly or inversely. $$ y=\frac{4}{x} $$

Step-by-Step Solution

Answer

The graph of the function \( y = \frac{4}{x} \) is a hyperbola suggesting that x and y are inversely related.

Step 1: Calculate y-values

For each x-value (-4, -3, -2, -1, 1, 2, 3, 4), substitute the x-value into the equation \( y = \frac{4}{x} \) and solve for y.

Step 2: Create a table

The calculated y-values in conjunction with their corresponding x-values can be represented in a table. This makes it easier to see the relationship between the x and y values.

Step 3: Sketch the graph

Plot the pairs of values on a graph. Since the sign of y changes as the sign of x changes, this indicates an inverse variation situation. The function curve should be a hyperbola.

Step 4: Identify the type of variation

Observe the resultant graph. Since y increases as x decreases and vice versa, we can conclude that x and y vary inversely.

Key Concepts

Graphing Inverse Functions

Graphing an inverse function like \( y = \frac{4}{x} \) involves understanding how changes in \( x \) affect \( y \). Start by identifying the key feature of an inverse function: unlike direct variation where \( y = kx \), an inverse function implies \( y = \frac{k}{x} \). This indicates that as \( x\) increases, \( y \) decreases and vice versa, reflecting inverse variation.
To plot this function, calculate \( y \)-values for chosen \( x \)-values. Substituting into the given equation, for example, \( x = -4 \) yields \( y = -1 \). Continue calculating for \( x = -3, -2, ..., 4 \) and collect these results to form ordered pairs.
Now plot these ordered pairs on the Cartesian plane. As you do, you'll notice a distinctive curved shape appearing. This graph suggests the function is a hyperbola, which we will explore further. Remember, since \( x \) and \( y \) are inversely related, the curve will never touch the axes, but it approaches them infinitely, forming a clear mirror-image symmetrical curve around the origin.

Hyperbola

A hyperbola is a type of smooth curve lying in a plane. It is identified by the shape created by functions like \( y = \frac{k}{x} \). With an inverse relationship, the graph consists of two separate branches, each located in adjacent quadrants of the coordinate plane.
In the case of \( y = \frac{4}{x} \), the branches of the hyperbola appear in the first and third quadrants when \( x \) and \( y \) are both positive or both negative. The curve's steepness and approach towards the axes depend on the constant \( k \) in \( \frac{k}{x} \).
Fascinatingly, a hyperbola represents a balance where the product of the coordinates is constant — in this case, the product is always 4. This consistent product further solidifies the hyperbolic nature, providing insightful visual context into inverse variation scenarios. These curving paths invite further exploration into symmetrical properties and transformations within the coordinate plane.

Creating Tables from Functions

Constructing a table of values is a helpful method for visualizing how \( x \) and \( y \) values interact in a given function. For the function \( y = \frac{4}{x} \), start by selecting a range of \( x \)-values. This exercise uses \( x = -4, -3, -2, ..., 4 \).
Plug each \( x \)-value into the function to solve for \( y \). For instance, with \( x = -3 \), calculate \( y = \frac{4}{-3} = -\frac{4}{3} \). Continue this process for each chosen \( x \)-value. Once all calculations are complete, organize them into a table.
This table serves as a roadmap for graphing. Each row pairs an \( x \)-value with its corresponding \( y \)-value, simplifying the plotting process. As patterns emerge, the nature of the variation becomes visible and aids in sketching the graph. It is through this organized reflection of data that inverse variations reveal their unique properties in a structured manner.