Before we delve into the transformations, let's first understand the original function f(x) = 4^x. This is an exponential function with a base of 4. Its key features are:...
- Always positive: The function always produces positive values, regardless of the input x. It never crosses the x-axis.
- Rapid growth: The function grows exponentially, meaning it increases very quickly as x increases. This is due to the base being greater than 1.
- Asymptotic behavior: As x approaches negative infinity, the function approaches 0. This means the x-axis serves as a horizontal asymptote for the function.
- Passing through (0, 1): When x = 0, f(x) = 4^0 = 1. The graph of the function always passes through the point (0, 1).
Reflection over the x-axis
Reflecting a function over the x-axis involves negating the entire function's output. This means we multiply the original function by -1. The transformed function becomes:
g(x) = -f(x) = -4^x
The reflection flips the graph over the x-axis. The key features of the original function are now reversed:
- Always negative: The function now produces negative values for all x values.
- Rapid decrease: The function decreases exponentially as x increases, due to the negative sign.
- Asymptotic behavior: As x approaches negative infinity, the function approaches 0 from below. The x-axis is still the horizontal asymptote.
- Passing through (0, -1): When x = 0, g(x) = -4^0 = -1. The graph passes through the point (0, -1).
Translation to the Right
Translating a function to the right involves shifting the graph horizontally. To translate a function 4 units to the right, we subtract 4 from the input x. The transformed function becomes:
h(x) = g(x - 4) = -4^(x - 4)
The translation shifts the graph 4 units to the right. This affects the key features as follows:
- Horizontal shift: All points on the graph shift 4 units to the right. The vertical asymptote also shifts 4 units to the right.
- No change in shape: The shape of the graph remains the same. It still represents an exponentially decreasing function.
- New x-intercept: The x-intercept shifts to the right by 4 units. The exact value of the new x-intercept can be found by setting h(x) = 0 and solving for x.
Understanding the Combined Transformation
After the reflection over the x-axis and the translation to the right, the final function becomes:
h(x) = -4^(x - 4)
The graph of this function is the same as the original f(x) = 4^x, but reflected over the x-axis and shifted 4 units to the right. This is a common approach in mathematics and other fields for transforming functions and analyzing their behavior under different conditions.
Visualizing the Transformations
The following steps illustrate the transformation process visually:
- Original Function: Start with the graph of f(x) = 4^x, a typical exponential function with rapid growth.
- Reflection: Reflect the graph over the x-axis, negating the output of the function. This flips the graph over the x-axis.
- Translation: Translate the reflected graph 4 units to the right by subtracting 4 from the input x. This shifts the graph horizontally to the right.
Key Points to Remember
- Reflecting a function over the x-axis involves multiplying the function by -1.
- Translating a function to the right involves subtracting a constant from the input x.
- Understanding the transformation steps helps visualize the changes in the graph and analyze the function's behavior.
- These transformations are fundamental in various mathematical fields and applications involving exponential functions.