Analyzing The Function H(x) = (2x - 6) / (x + 3) True Or False Statements

Let's delve into the intricacies of the function h(x) = (2x - 6) / (x + 3). This function, a rational function, presents a fascinating subject for mathematical exploration. To fully understand its behavior, we will dissect its key features, including its asymptotes, intercepts, and overall graph. This comprehensive analysis will not only reveal the function's unique characteristics but also provide a solid foundation for understanding other rational functions.

Asymptotes: The Invisible Boundaries

Asymptotes, the invisible lines that a function approaches but never quite touches, are crucial to understanding the function h(x) = (2x - 6) / (x + 3). They act as guides, shaping the graph's trajectory and revealing its behavior at extreme values of x. Let's explore the two types of asymptotes relevant to this function: vertical and horizontal.

Vertical Asymptotes: Where the Function Breaks

Vertical asymptotes occur where the denominator of a rational function equals zero. In our case, the denominator is (x + 3). Setting this equal to zero, we get x + 3 = 0, which solves to x = -3. This means that as x approaches -3, the function's value will either skyrocket towards positive infinity or plummet towards negative infinity. Therefore, there is a vertical asymptote at x = -3. The function is undefined at x = -3, creating a break in the graph. This vertical asymptote dictates the function's behavior near x = -3, preventing it from crossing this vertical line.

To further clarify, consider the values of x slightly less than -3 and slightly greater than -3. For x values slightly less than -3 (e.g., -3.1), the denominator (x + 3) becomes a small negative number, while the numerator (2x - 6) is negative. A negative number divided by a small negative number results in a large positive number, indicating that the function approaches positive infinity as x approaches -3 from the left. Conversely, for x values slightly greater than -3 (e.g., -2.9), the denominator (x + 3) becomes a small positive number, while the numerator remains negative. A negative number divided by a small positive number results in a large negative number, indicating that the function approaches negative infinity as x approaches -3 from the right. This behavior confirms the presence of a vertical asymptote at x = -3.

Horizontal Asymptotes: The Long-Term Trend

Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. In h(x) = (2x - 6) / (x + 3), both the numerator (2x - 6) and the denominator (x + 3) are linear polynomials, meaning they both have a degree of 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 2/1 = 2. This means that as x becomes very large (positive or negative), the function's value will approach 2, but it will never actually reach it.

To understand this intuitively, consider what happens as x becomes extremely large. The constant terms (-6 and +3) become insignificant compared to the x terms (2x and x). The function effectively behaves like (2x) / (x), which simplifies to 2. This confirms the presence of a horizontal asymptote at y = 2. The function will get closer and closer to this line as x moves towards infinity or negative infinity, but it will never cross it. This horizontal asymptote provides a crucial understanding of the function's long-term behavior.

Intercepts: Where the Function Meets the Axes

Intercepts, the points where the graph of the function intersects the x and y axes, provide valuable insights into the function's behavior. They help us understand where the function's value is zero (x-intercept) and what the function's value is when x is zero (y-intercept).

X-Intercept: The Zero Crossing

The x-intercept is the point where the graph crosses the x-axis. At this point, the y-value, or the function's value, is zero. To find the x-intercept, we set h(x) = 0 and solve for x:

0 = (2x - 6) / (x + 3)

For a fraction to be zero, the numerator must be zero. So, we set 2x - 6 = 0 and solve for x:

2x - 6 = 0

2x = 6

x = 3

Therefore, the x-intercept is at x = 3, which corresponds to the point (3, 0) on the graph. This point marks where the function's value changes from negative to positive or vice versa. The x-intercept is a critical point for understanding the function's overall shape and behavior.

Y-Intercept: The Starting Point

The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function h(x):

h(0) = (2(0) - 6) / (0 + 3)

h(0) = -6 / 3

h(0) = -2

Therefore, the y-intercept is at y = -2, which corresponds to the point (0, -2) on the graph. This point represents the function's initial value when x is zero. The y-intercept is often a convenient starting point for sketching the graph of the function.

Graphing h(x): Putting It All Together

Now that we've identified the asymptotes and intercepts of h(x) = (2x - 6) / (x + 3), we can sketch its graph. The asymptotes act as guides, and the intercepts provide key points on the graph.

1. Draw the Asymptotes: First, draw the vertical asymptote at x = -3 and the horizontal asymptote at y = 2. These lines divide the coordinate plane into regions and constrain the function's behavior.
2. Plot the Intercepts: Plot the x-intercept at (3, 0) and the y-intercept at (0, -2). These points anchor the graph and help define its shape.
3. Sketch the Curve: Now, sketch the curve, keeping in mind the asymptotes and intercepts. The function will approach the asymptotes as x approaches infinity or -3. Since the function approaches negative infinity as x approaches -3 from the right and the y-intercept is (0, -2), the graph will start in the third quadrant, pass through (0, -2), and then approach the vertical asymptote x = -3. On the other side of the vertical asymptote, since the function approaches positive infinity as x approaches -3 from the left and we have the x-intercept at (3,0), the graph will start near the top of the second quadrant, and pass through the x-intercept and then approach the horizontal asymptote y=2 as x gets larger. The horizontal asymptote y = 2 is a horizontal boundary, meaning the function will get closer and closer to y = 2 as x gets more positive or negative, but it won’t cross the horizontal asymptote.

By combining our understanding of asymptotes and intercepts, we can create a reasonably accurate sketch of the graph of h(x) = (2x - 6) / (x + 3). This visual representation further enhances our comprehension of the function's behavior.

Conclusion: A Complete Picture

By meticulously analyzing the asymptotes, intercepts, and overall behavior of the function h(x) = (2x - 6) / (x + 3), we've gained a comprehensive understanding of its characteristics. The vertical asymptote at x = -3 and the horizontal asymptote at y = 2 define the function's boundaries, while the x-intercept at (3, 0) and the y-intercept at (0, -2) provide key points on the graph. This detailed analysis serves as a powerful example of how we can dissect and understand rational functions, providing a valuable toolkit for tackling more complex mathematical challenges. This process illuminates the importance of understanding asymptotes and intercepts in grasping the nature of functions and their graphical representations.

Determining True or False Statements About the Graph of h(x)

Now, let's address the core question: determining the truthfulness of various statements about the graph of h(x) = (2x - 6) / (x + 3). Based on our comprehensive analysis, we can evaluate the accuracy of different assertions.

To effectively address this, we need specific statements to analyze. However, based on our thorough examination of the function, we can anticipate some common types of statements and how to evaluate them.

Here are some examples of statements and how we would determine their truthfulness:

• Statement: The graph of h(x) has a vertical asymptote at x = -3.
  • Truthfulness: True. We determined this earlier by finding the zero of the denominator.
• Statement: The graph of h(x) has a horizontal asymptote at y = 2.
  • Truthfulness: True. We found this by comparing the degrees of the numerator and denominator.
• Statement: The graph of h(x) intersects the x-axis at x = 3.
  • Truthfulness: True. This is our calculated x-intercept.
• Statement: The graph of h(x) intersects the y-axis at y = -2.
  • Truthfulness: True. This is our calculated y-intercept.
• Statement: The graph of h(x) passes through the point (1, -1).
  • Truthfulness: To verify this, we substitute x = 1 into the function:
    • h(1) = (2(1) - 6) / (1 + 3) = -4 / 4 = -1. True.
• Statement: The function h(x) is always increasing.
  • Truthfulness: To assess this, we would typically analyze the function's derivative. However, based on the graph's general shape, we can infer that this statement is False. The function increases on the intervals (-infinity, -3) and (-3, infinity), but it is not increasing across x=-3 due to the discontinuity.
• Statement: The range of h(x) is all real numbers.
  • Truthfulness: False. The horizontal asymptote at y = 2 indicates that the function will never actually reach the value of 2. Therefore, 2 is not in the range of the function. The range is all real numbers except 2.

In general, to determine the truthfulness of statements about the graph of h(x), we can use the following strategies:

• Asymptotes: Verify statements about vertical asymptotes by checking the zeros of the denominator. Verify statements about horizontal asymptotes by comparing the degrees of the numerator and denominator.
• Intercepts: Verify statements about x-intercepts by setting h(x) = 0 and solving for x. Verify statements about y-intercepts by evaluating h(0).
• Specific Points: To check if the graph passes through a specific point (a, b), substitute x = a into h(x) and see if the result is b.
• Increasing/Decreasing Behavior: A more rigorous analysis would involve calculus (finding the derivative). However, a general understanding can be gleaned from the graph's shape.
• Range: Consider the horizontal asymptotes and any restrictions on the function's output to determine the range.

By systematically applying these techniques, we can confidently assess the truthfulness of any statement about the graph of h(x) = (2x - 6) / (x + 3).

This detailed exploration equips you with the tools and knowledge necessary to confidently analyze rational functions and their graphical representations. Remember to focus on the key features – asymptotes, intercepts, and overall behavior – to unlock the secrets hidden within these fascinating mathematical entities.