Decreasing Intervals Of Quadratic Function F(x)

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Decreasing Intervals of Quadratic Function f(x)

Let's dive into finding the intervals where the given quadratic function f(x) = -2(x - 1)^2 + 8 is decreasing. This is a classic problem in algebra that combines understanding quadratic functions with calculus concepts like increasing and decreasing intervals. So, buckle up, and let’s get started!

Understanding the Quadratic Function

Before we jump into finding where the function is decreasing, let's break down what the function f(x) = -2(x - 1)^2 + 8 actually represents. This is a quadratic function expressed in vertex form, which is super handy for identifying key features of the parabola it describes. The general vertex form of a quadratic function is given by:

f(x) = a(x - h)^2 + k

Where:

  • (h, k) is the vertex of the parabola.
  • a determines whether the parabola opens upwards (a > 0) or downwards (a < 0), and also affects the “width” of the parabola.

In our case, f(x) = -2(x - 1)^2 + 8, we can identify:

  • a = -2
  • h = 1
  • k = 8

This tells us that the vertex of our parabola is at the point (1, 8). Also, since a = -2 is negative, the parabola opens downwards. This is crucial because a downward-opening parabola increases up to its vertex and then decreases afterward. Knowing this shape helps us anticipate where the function will be decreasing.

The coefficient a = -2 also tells us that the parabola is vertically stretched by a factor of 2 and reflected across the x-axis. This means it's narrower and opens downward compared to the standard parabola y = x^2. Understanding these transformations helps in visualizing the graph and predicting its behavior.

To summarize, by recognizing the vertex form of the quadratic function, we've quickly identified the vertex (1, 8) and the direction of the parabola (downward). This sets the stage for determining the intervals where the function is decreasing. Remember, the vertex is the turning point, and for a downward-opening parabola, the function decreases to the right of the vertex.

Determining Decreasing Intervals

Now, let's pinpoint exactly where the function f(x) = -2(x - 1)^2 + 8 is decreasing. Since we know the parabola opens downwards and its vertex is at (1, 8), we can deduce that the function increases up to x = 1 and then decreases for all x > 1. This is because, to the left of the vertex, the y-values are increasing as x increases, and to the right of the vertex, the y-values are decreasing as x increases.

In mathematical terms, the function is decreasing on the interval (1, ∞). This notation means that the function decreases for all values of x greater than 1, extending indefinitely to positive infinity. Note that we use an open interval at x = 1 because, at the vertex itself, the function is neither increasing nor decreasing; it's at a critical point where the slope momentarily equals zero.

To further clarify, consider what happens as x moves to the right of 1. For example:

  • At x = 2, f(2) = -2(2 - 1)^2 + 8 = -2(1) + 8 = 6
  • At x = 3, f(3) = -2(3 - 1)^2 + 8 = -2(4) + 8 = 0
  • At x = 4, f(4) = -2(4 - 1)^2 + 8 = -2(9) + 8 = -10

As you can see, the value of f(x) decreases as x increases beyond 1. This confirms our deduction that the function is decreasing on the interval (1, ∞). It's crucial to understand this behavior by visualizing the graph. Imagine starting at the vertex (1, 8) and moving to the right along the curve; the y-values continuously decrease.

So, the key takeaway here is that for a downward-opening parabola, the function decreases on the interval to the right of the vertex. Identifying the vertex from the vertex form of the quadratic function allows us to quickly determine this decreasing interval. This method provides a straightforward and efficient way to solve this type of problem.

Alternative Approach: Using Calculus

For those familiar with calculus, we can also use derivatives to find where the function f(x) = -2(x - 1)^2 + 8 is decreasing. The derivative of a function gives us the slope of the tangent line at any point, and where the derivative is negative, the function is decreasing.

First, let's find the derivative of f(x):

f(x) = -2(x - 1)^2 + 8

f(x) = -2(x^2 - 2x + 1) + 8

f(x) = -2x^2 + 4x - 2 + 8

f(x) = -2x^2 + 4x + 6

Now, we find the derivative f'(x):

f'(x) = -4x + 4

To find where the function is decreasing, we need to solve for f'(x) < 0:

-4x + 4 < 0

-4x < -4

x > 1

This result tells us that the function is decreasing for all x > 1, which matches our earlier conclusion using the properties of the vertex form. The interval where f(x) is decreasing is therefore (1, ∞).

Using calculus provides a more rigorous approach. By finding the derivative and determining where it is negative, we can precisely identify the intervals where the original function is decreasing. This method is especially useful for more complex functions where identifying the vertex isn't straightforward.

In summary, the calculus approach confirms our previous finding: the function f(x) = -2(x - 1)^2 + 8 is decreasing on the interval (1, ∞). This method reinforces the relationship between the derivative and the increasing/decreasing behavior of a function. Whether you prefer using properties of the vertex form or applying calculus, understanding both approaches provides a comprehensive grasp of the problem.

Graphical Interpretation

Another helpful way to understand where the function f(x) = -2(x - 1)^2 + 8 is decreasing is through a graphical interpretation. Visualizing the graph of the function can provide an intuitive understanding of its behavior. The graph of f(x) is a parabola that opens downwards, with its vertex at the point (1, 8).

Imagine plotting the graph on a coordinate plane. As you move from left to right along the x-axis, observe how the y-values change. To the left of the vertex (i.e., for x < 1), the y-values increase as x increases. This means the function is increasing in this interval. At the vertex (1, 8), the function reaches its maximum value.

Now, focus on what happens to the right of the vertex (i.e., for x > 1). As you continue to move from left to right, you'll notice that the y-values decrease as x increases. This indicates that the function is decreasing in this interval. The rate of decrease becomes steeper as you move further away from the vertex.

The graph visually confirms that the function is decreasing on the interval (1, ∞). It provides a clear picture of how the function's values change as x moves away from the vertex towards positive infinity. This graphical perspective is particularly useful for those who find visual aids helpful in understanding mathematical concepts.

Moreover, the graph highlights the symmetry of the parabola around the vertical line x = 1. While the function is decreasing on (1, ∞), it's increasing on (-∞, 1). This symmetry is a characteristic feature of quadratic functions and can help in quickly identifying intervals of increasing and decreasing behavior.

In conclusion, the graphical interpretation offers a valuable supplement to the analytical methods we've discussed. By visualizing the graph of f(x) = -2(x - 1)^2 + 8, we can intuitively understand why the function is decreasing on the interval (1, ∞). This approach reinforces the concept and provides a more complete understanding of the function's behavior.

Conclusion

In summary, we've explored various methods to determine the interval where the quadratic function f(x) = -2(x - 1)^2 + 8 is decreasing. By recognizing the vertex form of the quadratic function, we identified the vertex at (1, 8) and deduced that the parabola opens downwards. This immediately told us that the function decreases to the right of the vertex.

We then confirmed this finding using calculus by taking the derivative of the function and solving for where f'(x) < 0. This approach provided a more rigorous mathematical justification for our conclusion. Finally, we used a graphical interpretation to visualize the behavior of the function, reinforcing our understanding that the function is decreasing on the interval (1, ∞).

Each of these methods – using the vertex form, applying calculus, and interpreting the graph – provides a different perspective on the problem. Understanding all three approaches can deepen your comprehension of quadratic functions and their behavior. Whether you prefer algebraic manipulation, calculus techniques, or visual aids, having multiple tools at your disposal can help you tackle similar problems with confidence.

Therefore, the definitive answer is that the function f(x) = -2(x - 1)^2 + 8 is decreasing on the interval (1, ∞). This comprehensive analysis should give you a solid understanding of how to approach such problems in the future. Keep practicing and exploring different methods to enhance your problem-solving skills!