(x^2-7x+10)/(x+3)

4 min read Jun 17, 2024

Exploring the Expression (x^2 - 7x + 10)/(x+3)

This expression represents a rational function, which is a function formed by dividing one polynomial by another. Let's break down its key features and explore its behavior.

Factoring and Simplifying

First, we can factor the numerator:

(x^2 - 7x + 10) = (x - 5)(x - 2)

Therefore, our expression becomes:

(x - 5)(x - 2) / (x + 3)

This factored form reveals potential points of interest:

Vertical Asymptote: The denominator becomes zero when x = -3. This indicates a vertical asymptote at x = -3. The function approaches infinity as x approaches -3.
Holes: There are no common factors between the numerator and denominator after simplification, so there are no holes in the graph.
X-Intercepts: The numerator becomes zero when x = 5 and x = 2. These are the x-intercepts of the function.
Y-Intercept: To find the y-intercept, we set x = 0: (0 - 5)(0 - 2)/(0 + 3) = 10/3. The y-intercept is at (0, 10/3).

Analyzing the Function's Behavior

As x approaches positive infinity: The function approaches a horizontal asymptote at y = 1. This is because the degree of the numerator and denominator are the same, and the leading coefficients are both 1.
As x approaches negative infinity: The function also approaches the same horizontal asymptote, y = 1.

Graphing the Function

To visualize the function, consider the following:

Plot the intercepts: Plot the points (5, 0), (2, 0), and (0, 10/3).
Draw the vertical asymptote: Draw a vertical line at x = -3.
Draw the horizontal asymptote: Draw a horizontal line at y = 1.
Sketch the curve: Connect the intercepts and approach the asymptotes. Remember that the function cannot cross the vertical asymptote.

The graph will show a curve with two branches, one on each side of the vertical asymptote, approaching the horizontal asymptote as x approaches positive and negative infinity.

Conclusion

Understanding the factored form and analyzing the function's behavior allows us to effectively visualize and interpret the function (x^2 - 7x + 10)/(x+3). This process helps us understand the key features of rational functions and their graphical representations.