Graphing F(x) = -(x - 2)³ - 5 Using Transformations Domain And Range

by Admin 69 views

In this comprehensive guide, we will explore the process of graphing the function f(x) = -(x - 2)³ - 5 using transformations. Understanding transformations is crucial for visualizing and analyzing functions effectively. We will break down the function into its constituent transformations, graph each transformation step-by-step, and ultimately arrive at the graph of the given function. Additionally, we will delve into determining the domain and range of the function, providing a complete understanding of its behavior.

Understanding Function Transformations

Before we dive into graphing the function, let's recap the fundamental concepts of function transformations. Transformations involve altering the basic form of a function to shift, stretch, compress, or reflect its graph. The key transformations we'll focus on are:

  • Vertical Shifts: Adding or subtracting a constant from the function shifts the graph vertically. f(x) + c shifts the graph upward by c units, while f(x) - c shifts it downward by c units.
  • Horizontal Shifts: Replacing x with (x - c) shifts the graph horizontally. f(x - c) shifts the graph to the right by c units, while f(x + c) shifts it to the left by c units.
  • Reflections: Multiplying the function by -1 reflects the graph across the x-axis. -f(x) reflects the graph vertically. Similarly, replacing x with -x reflects the graph across the y-axis. f(-x) reflects the graph horizontally.
  • Vertical Stretches and Compressions: Multiplying the function by a constant a stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. af(x)* stretches the graph vertically by a factor of a if a > 1, compresses it if 0 < a < 1, reflects it across the x-axis if a < 0, and does a combination of stretching/compressing and reflecting if a is negative and not equal to -1.
  • Horizontal Stretches and Compressions: Replacing x with bx stretches the graph horizontally if 0 < |b| < 1 and compresses it if |b| > 1. f(bx) compresses the graph horizontally by a factor of b if b > 1, stretches it if 0 < b < 1, and may also involve reflection across the y-axis if b is negative.

Step-by-Step Graphing of f(x) = -(x - 2)³ - 5

Now, let's apply these transformations to graph f(x) = -(x - 2)³ - 5. We'll start with the basic cubic function, g(x) = x³, and apply transformations sequentially.

1. The Basic Cubic Function: g(x) = x³

The graph of g(x) = x³ is a classic cubic curve that passes through the origin (0, 0). It increases from left to right, with a point of inflection at the origin. The basic cubic function serves as the foundation for our transformations. The key points to visualize on this graph are (-1, -1), (0, 0), and (1, 1).

2. Horizontal Shift: g₁(x) = (x - 2)³

Next, we apply a horizontal shift. The transformation (x - 2) indicates a shift to the right by 2 units. So, g₁(x) = (x - 2)³ is the graph of g(x) = x³ shifted 2 units to the right. The point (0, 0) on the original graph moves to (2, 0), and the other key points shift accordingly: (-1, -1) becomes (1, -1), and (1, 1) becomes (3, 1). This shift changes the function's x-intercept and repositions the entire curve along the x-axis.

3. Reflection Across the x-axis: g₂(x) = -(x - 2)³

Now, we reflect the graph across the x-axis. The negative sign in front of the function, -, indicates this reflection. Thus, g₂(x) = -(x - 2)³ is the reflection of g₁(x) = (x - 2)³ across the x-axis. This transformation inverts the graph vertically. For example, the point (2, 0) remains unchanged (since it's on the x-axis), but (3, 1) becomes (3, -1), and (1, -1) becomes (1, 1). This reflection fundamentally alters the increasing/decreasing behavior of the function.

4. Vertical Shift: f(x) = -(x - 2)³ - 5

Finally, we apply a vertical shift. The - 5 at the end of the function indicates a shift downward by 5 units. Therefore, f(x) = -(x - 2)³ - 5 is the graph of g₂(x) = -(x - 2)³ shifted 5 units downward. This transformation moves every point on the graph down by 5 units. For instance, (2, 0) moves to (2, -5), (3, -1) moves to (3, -6), and (1, 1) moves to (1, -4). This final shift positions the graph in its ultimate location on the coordinate plane.

By sequentially applying these transformations – the horizontal shift, reflection across the x-axis, and vertical shift – we have successfully graphed the function f(x) = -(x - 2)³ - 5. Each step builds upon the previous one, illustrating how transformations can be used to construct complex graphs from simpler ones.

Determining the Domain of f(x) = -(x - 2)³ - 5

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To find the domain of f(x) = -(x - 2)³ - 5, we need to consider any restrictions on the input values.

In this case, the function is a cubic polynomial. Polynomial functions are defined for all real numbers, meaning there are no restrictions on the input values. We can substitute any real number for x, and the function will produce a real number output. There are no denominators that could be zero, no square roots of negative numbers, or other restrictions that would limit the domain.

Therefore, the domain of f(x) = -(x - 2)³ - 5 is all real numbers.

In interval notation, the domain is expressed as (-∞, ∞), indicating that the function is defined for all x-values from negative infinity to positive infinity.

Determining the Range of f(x) = -(x - 2)³ - 5

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range of f(x) = -(x - 2)³ - 5, we analyze the function's behavior as x varies over its domain.

Since the function is a cubic polynomial, it will take on all real number values. The cubic term, (x - 2)³, can become infinitely large in both the positive and negative directions as x varies. The reflection across the x-axis (the negative sign) does not limit the range; it simply inverts the graph. Similarly, the vertical shift of -5 just moves the entire range down by 5 units but does not change the fact that the function can still take on all real values.

As x approaches positive infinity, -(x - 2)³ approaches negative infinity, and therefore f(x) approaches negative infinity. Conversely, as x approaches negative infinity, -(x - 2)³ approaches positive infinity, and therefore f(x) approaches positive infinity.

Thus, the range of f(x) = -(x - 2)³ - 5 is all real numbers.

In interval notation, the range is expressed as (-∞, ∞), indicating that the function can produce any real number as an output.

Conclusion

In this guide, we've thoroughly examined the process of graphing the function f(x) = -(x - 2)³ - 5 using transformations. By breaking the function down into sequential transformations – a horizontal shift, reflection across the x-axis, and a vertical shift – we were able to construct the graph step-by-step. Additionally, we determined that both the domain and the range of the function are all real numbers, (-∞, ∞). Understanding transformations and how they affect the graph of a function is a powerful tool in mathematical analysis and visualization.

By mastering these techniques, you can confidently analyze and graph a wide variety of functions, gaining deeper insights into their behavior and properties. The combination of graphical transformations and domain/range determination provides a comprehensive understanding of function analysis.