Problem 56 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe the intervals where the graph of \(y=\frac{a}{x}\) is increasing or decreasing when (a) \(a>0\) and (b) \(a<0\). Explain your reasoning.

Step-by-Step Solution

Answer

(a) When \(a>0\), the function decreases on the interval \((-\infty ,0)\) and increases on the interval \((0,+\infty)\). (b) When \(a<0\) the function increases on the interval \((-\infty,0)\) and decreases on the interval \((0,+\infty)\).

Step 1: Analyze the function when \(a>0\)

When \(a\) is a positive number, the function \(y=\frac{a}{x}\) will be positive for \(x>0\) and negative for \(x<0\). Therefore, the function decreases for \(x<0\) as its value changes from positive to negative. And the function increases for \(x>0\) as its value goes from negative to positive.

Step 2: Analyze the function when \(a<0\)

When \(a\) is a negative number, the function \(y=\frac{a}{x}\) turns out to be negative for \(x>0\) and positive for \(x<0\). Therefore, the function decreases for \(x>0\) and increases for \(x<0\) as its value changes from positive to negative and from negative to positive respectively.

Key Concepts

Increasing Intervals

In mathematics, an increasing interval of a function is where the output of the function becomes larger as the input increases. For the rational function \(y = \frac{a}{x}\), understanding when the function is increasing depends largely on the sign of \(a\).

- **When \(a > 0\):** For positive values of \(a\), the function decreases when \(x < 0\) and increases when \(x > 0\). This is because as \(x\) goes from negative to positive, the fraction \(\frac{a}{x}\) goes from negative values to positive values, indicating an increasing interval for \(x > 0\).

- **When \(a < 0\):** In contrast, if \(a\) is negative, the function \(y = \frac{a}{x}\) is positive for \(x < 0\) and becomes negative for \(x > 0\). Therefore, the function increases on the interval when \(x < 0\), as the values go from a more negative towards less negative or positive as \(x\) decreases in the negative direction.

Recognizing these increasing intervals can help in sketching graphs and understanding the behavior of rational functions in various contexts.

Decreasing Intervals

Decreasing intervals of a function occur where the output of the function gets smaller as the input increases. For the function \(y = \frac{a}{x}\), let's explore when this happens based on the value of \(a\).

- **When \(a > 0\):** For \(a > 0\), the function is decreasing on the interval of \(x < 0\). As \(x\) moves closer to zero, from negative values, \(y = \frac{a}{x}\) transitions from positive values (since a positive divided by a negative is a negative) towards zero, hence decreasing.

- **When \(a < 0\):** Conversely, for \(a < 0\), the rational function is decreasing for \(x > 0\). As \(x\) increases, \(y = \frac{a}{x}\) changes from larger to smaller negative values, showing a decrease.

Understanding where a function decreases helps in evaluating the slope of the function curve and predicting the graphical behavior across different domains.

Asymptotic Behavior

The asymptotic behavior in rational functions like \(y = \frac{a}{x}\) refers to how the function behaves as \(x\) approaches certain critical points—such as zero or infinity—where the function is not defined.

**Vertical Asymptotes:** The function \(y = \frac{a}{x}\) has a vertical asymptote at \(x = 0\). As \(x\) approaches zero from either the positive or negative side, the function value \(y\) becomes infinitely large or infinitely small, respectively. This indicates that the graph will approach but never actually touch the line \(x = 0\).

**Horizontal Asymptotes:** As \(x\) approaches infinity or negative infinity, the value of \(\frac{a}{x}\) approaches zero. This suggests that the horizontal line \(y = 0\) is an asymptote that the graph gets closer to but never crosses as \(x\) becomes very large or very small.

Recognizing the asymptotic behavior is key to sketching the function's graph accurately and understanding how it behaves at the limits of its domain.