Zeros And End Behavior Of F(x) = X^3 - 8x^2 + X + 42
Let's dive into the fascinating world of polynomial functions! In this article, we're going to explore a specific cubic function, f(x) = x³ - 8x² + x + 42. Our journey will involve determining if certain values are zeros of the function and describing the function's end behavior, all without relying on fancy technology. So, grab your thinking caps, and let's get started!
Part A: Verifying the Zeros of f(x)
The first part of our exploration focuses on identifying the zeros of the function. Specifically, we want to know if f(x) has zeros located at 7, -2, and 3. A zero of a function is simply a value of x that makes the function equal to zero, i.e., f(x) = 0. To verify whether these values are indeed zeros, we'll use the direct substitution method. This involves plugging each value into the function and checking if the result is zero.
Testing x = 7
Let's start with x = 7. We substitute this value into our function:
f(7) = (7)³ - 8(7)² + (7) + 42
Now, let's break down the calculation step by step:
- (7)³ = 7 * 7 * 7 = 343
- 8(7)² = 8 * 49 = 392
Substituting these back into the equation, we get:
f(7) = 343 - 392 + 7 + 42
Combining the terms:
f(7) = 343 + 7 + 42 - 392 = 392 - 392 = 0
Since f(7) = 0, we can confidently conclude that x = 7 is a zero of the function f(x). This means that the graph of the function crosses the x-axis at the point (7, 0).
Testing x = -2
Next, let's test x = -2:
f(-2) = (-2)³ - 8(-2)² + (-2) + 42
Again, let's break down the calculation:
- (-2)³ = -2 * -2 * -2 = -8
- 8(-2)² = 8 * 4 = 32
Substituting these back into the equation, we have:
f(-2) = -8 - 32 - 2 + 42
Combining the terms:
f(-2) = -42 + 42 = 0
As f(-2) = 0, we confirm that x = -2 is also a zero of the function f(x). The graph of the function intersects the x-axis at the point (-2, 0).
Testing x = 3
Finally, let's check x = 3:
f(3) = (3)³ - 8(3)² + (3) + 42
Breaking down the calculation:
- (3)³ = 3 * 3 * 3 = 27
- 8(3)² = 8 * 9 = 72
Substituting these back into the equation:
f(3) = 27 - 72 + 3 + 42
Combining the terms:
f(3) = 27 + 3 + 42 - 72 = 72 - 72 = 0
Since f(3) = 0, we can definitively say that x = 3 is a zero of the function f(x). This point (3, 0) is where the graph of the function crosses the x-axis.
Conclusion for Part A
Through direct substitution and careful calculation, we have successfully verified that f(x) indeed has zeros located at x = 7, x = -2, and x = 3. This means that these three values are the points where the graph of the function intersects the x-axis. The zeros of a function are crucial for understanding its behavior and for graphing it accurately. Knowing these zeros, we can now move on to describing the end behavior of the function.
Part B: Describing the End Behavior of f(x)
The end behavior of a function describes what happens to the function's values (the y-values) as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). In simpler terms, we want to know where the graph of the function is heading as we move far to the right and far to the left on the x-axis. For polynomial functions, the end behavior is primarily determined by the leading term, which is the term with the highest power of x. In our case, the leading term of f(x) = x³ - 8x² + x + 42 is x³.
Understanding the Leading Term
The leading term, x³, is a cubic term with a positive coefficient (1). This gives us key information about the end behavior. The degree of the term (3) is odd, and the coefficient is positive. Let's break down how these factors influence the end behavior:
- Odd Degree: Odd-degree polynomials have opposite end behaviors. This means that as x approaches positive infinity, y will go in one direction, and as x approaches negative infinity, y will go in the opposite direction.
- Positive Coefficient: A positive leading coefficient means that as x approaches positive infinity, y will also approach positive infinity. This is because as x gets very large, x³ becomes a very large positive number.
Describing the End Behavior in Detail
Now, let's put it all together to describe the end behavior of f(x):
- As x approaches positive infinity (x → ∞): Since the leading coefficient is positive and the degree is odd, the function will also approach positive infinity (f(x) → ∞). This means that as we move to the right on the graph, the y-values will increase without bound.
- As x approaches negative infinity (x → -∞): Because the degree is odd and the coefficient is positive, the function will approach negative infinity (f(x) → -∞). This means that as we move to the left on the graph, the y-values will decrease without bound.
Visualizing the End Behavior
Imagine the graph of f(x). As you move far to the right along the x-axis, the graph shoots upwards towards positive infinity. Conversely, as you move far to the left along the x-axis, the graph plunges downwards towards negative infinity. This is the characteristic end behavior of a cubic function with a positive leading coefficient.
Conclusion for Part B
In conclusion, the end behavior of f(x) = x³ - 8x² + x + 42 can be described as follows:
- As x → ∞, f(x) → ∞
- As x → -∞, f(x) → -∞
This description provides a clear picture of how the function behaves at the extremes of its domain. Understanding end behavior is crucial for sketching the graph of a polynomial function and for analyzing its overall characteristics.
Wrapping Up
In this article, we've taken a comprehensive look at the function f(x) = x³ - 8x² + x + 42. We successfully verified that x = 7, x = -2, and x = 3 are zeros of the function using direct substitution. Furthermore, we described the end behavior of the function, explaining how it approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. These insights provide a solid understanding of the function's key properties.
To deepen your understanding of polynomial functions, consider exploring additional resources such as Khan Academy's Polynomial Functions Section. Happy learning!
