The expression (3x + 2)/(x - 5) is a rational expression that represents the quotient of two polynomials: the numerator, 3x + 2, and the denominator, x - 5. This type of expression is crucial for understanding how to manipulate and solve equations involving fractions, especially in contexts where simplification, finding limits, or solving for variable values is necessary. Rational expressions can exhibit properties such as asymptotes and holes, which become particularly relevant when analyzing their behavior in different scenarios.
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To simplify (3x + 2)/(x - 5), you check for common factors in the numerator and denominator. In this case, there are none to cancel out.
The expression has a vertical asymptote at x = 5 since the denominator becomes zero, causing the expression to be undefined.
When finding the intercepts of the rational expression, set the numerator equal to zero; for (3x + 2), this gives an x-intercept at x = -2/3.
For large values of x, the behavior of (3x + 2)/(x - 5) approaches the horizontal asymptote y = 3 because the degrees of the numerator and denominator are equal.
When graphing (3x + 2)/(x - 5), it is important to note where it is undefined and how it behaves near its vertical asymptote.
Review Questions
How would you identify and analyze the vertical asymptote of the rational expression (3x + 2)/(x - 5)?
To identify the vertical asymptote of (3x + 2)/(x - 5), you set the denominator equal to zero. Solving x - 5 = 0 gives x = 5 as the location of the asymptote. This means that as x approaches 5 from either side, the value of the expression will tend toward positive or negative infinity, indicating that there’s a discontinuity at this point on the graph.
What is the significance of finding intercepts in relation to the expression (3x + 2)/(x - 5), and how would you calculate them?
Finding intercepts is significant as it helps understand where the graph crosses the axes. For the x-intercept, set the numerator equal to zero: 3x + 2 = 0 leads to x = -2/3. The y-intercept occurs when x = 0: substituting gives (3(0) + 2)/(0 - 5) = -2/5. These points are critical for sketching the graph of the rational expression.
Evaluate how understanding the behavior of (3x + 2)/(x - 5) as x approaches infinity can help predict its long-term trends.
As x approaches infinity, we observe that (3x + 2)/(x - 5) simplifies toward its leading coefficients because lower degree terms become insignificant. This means we look at y = 3/x which trends towards y = 3 as x gets very large. This insight helps predict that regardless of specific values, the function stabilizes around this horizontal asymptote for large inputs, indicating that it will not grow indefinitely but rather level off at y = 3.
Related terms
Rational Function: A function that can be expressed as the ratio of two polynomial functions.
Asymptote: A line that a graph approaches but never touches; it can be vertical, horizontal, or oblique.
Simplification: The process of reducing a rational expression to its simplest form by factoring and canceling common factors.