Problem 20 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=\frac{x^{2}-1}{x-1}\)

Step-by-Step Solution

Answer

The function is linear with a hole at \( x = 1 \).

Step 1: Identify the Form of the Function

Begin by simplifying the given equation, if possible. The equation is \( y = \frac{x^2 - 1}{x - 1} \). Factor the numerator as \( x^2 - 1 = (x - 1)(x + 1) \). So the equation becomes \( y = \frac{(x - 1)(x + 1)}{x - 1} \).

Step 2: Simplify the Equation

Since we have \( \frac{(x - 1)(x + 1)}{x - 1} \), cancel the \( x - 1 \) from the numerator and denominator. This results in \( y = x + 1 \) for \( x eq 1 \).

Step 3: Identify the Function Type

The simplified equation, \( y = x + 1 \), is in the form of a linear function \( y = mx + b \), where \( m = 1 \) and \( b = 1 \). This means the original equation represents a linear function.

Step 4: Considerations for Graphing

While the equation simplifies to \( y = x + 1 \), remember the original function had a restriction. At \( x = 1 \), the function is undefined. Hence, the graph of \( y = x + 1 \) will have a hole at \( x = 1 \).

Step 5: Graph the Equation

To graph this function, draw the line \( y = x + 1 \). Mark a hole at the point \( (1, 2) \) because the function is undefined at \( x = 1 \). The rest of the line follows the linear path with a slope of 1 and intercepts the y-axis at 1.

Key Concepts

Linear Functions

Linear functions are one of the most foundational types of functions in mathematics. These functions can be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For example, the function \( y = x + 1 \) shows a slope \( m = 1 \) and a y-intercept \( b = 1 \).

Linear functions graph as straight lines, with the slope determining the steepness. A positive slope means the line goes upwards, while a negative slope sends it downwards. The y-intercept is where the line crosses the y-axis. In our example, the line crosses at the point \( (0, 1) \).

Understanding linear functions is crucial because they form the building blocks for more complex mathematical concepts.

Factoring Polynomials

Factoring polynomials is a key skill that simplifies complex expressions. It involves expressing polynomials as the product of their factors. For instance, the polynomial \( x^2 - 1 \) can be factored as \( (x - 1)(x + 1) \).

This factorization is known as the difference of squares, which follows the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a \) is \( x \) and \( b \) is 1.

Factoring is particularly useful when simplifying rational functions and can help in canceling out terms to reveal the underlying structure, as seen in the process that converted \( \frac{x^2 - 1}{x - 1} \) into a simple linear function \( x + 1 \).

Practicing factoring polynomials can greatly aid in solving algebraic equations and understanding polynomial behaviors.

Undefined Points

Undefined points occur in functions where certain values make the function impossible to evaluate. For rational functions, these are typically where the denominator is zero.

In the example \( \frac{x^2 - 1}{x - 1} \), factoring gives \( \frac{(x-1)(x+1)}{x-1} \). By canceling \( x - 1 \), the function seems to become \( y = x + 1 \). However, at \( x = 1 \), the original function's denominator is zero, so \( y \) is undefined.

This results in a hole in the graph at \( x = 1 \). So, while the line \( y = x + 1 \) is drawn, there is an open circle at \( (1, 2) \) to indicate the point is not included.

Understanding undefined points is essential as they often indicate crucial function behavior that impacts the overall graph.

Graphing Equations

Graphing equations visually represents functions on a coordinate plane, providing insights into their behavior. For linear functions like \( y = x + 1 \), graphing involves drawing a straight line through the y-intercept and using the slope to determine other points.

For our function, it starts at \( (0, 1) \) and moves up one unit and right one unit repetitively due to its slope of 1. However, the point \( (1, 2) \) is a hole, displayed with a small open circle, as the function is undefined there.

Accurate graphing requires careful plotting of key points, recognition of undefined points, and consideration of the line’s direction based on the slope. With practice, graphing can become a straightforward process to quickly understand complex functions. Understanding how graphs relate to equations helps to visualize how mathematical models work in real-world situations.