Finding G(x) After Transformations Of F(x) = 4√(x)

Understanding the Transformation of Functions

In mathematics, function transformations play a crucial role in understanding how the graph of a function changes when certain operations are applied. These operations can include shifts (translations), stretches, compressions (scalings), and reflections. By understanding these transformations, we can easily manipulate and analyze functions, which is fundamental in various fields such as physics, engineering, and computer science.

When dealing with transformations, it's essential to grasp the effects of horizontal and vertical shifts. A horizontal shift moves the graph left or right along the x-axis, while a vertical shift moves the graph up or down along the y-axis. Understanding these shifts is critical for grasping function behavior and predicting their graphical representations. A shift to the right means we are subtracting a constant from $x$, and a shift to the left means we are adding a constant to $x$. Conversely, a vertical shift upwards involves adding a constant to the entire function, and a shift downwards involves subtracting a constant.

In addition to shifts, functions can also undergo stretches and compressions. Vertical stretches increase the y-values, making the graph taller, whereas vertical compressions decrease the y-values, making the graph shorter. Similarly, horizontal stretches widen the graph, and horizontal compressions narrow it. Recognizing these transformations is key to quickly sketching and analyzing functions.

Reflections are another essential type of transformation. A reflection across the x-axis flips the graph over the x-axis, effectively changing the sign of the y-values. Similarly, a reflection across the y-axis flips the graph over the y-axis, changing the sign of the x-values. The ability to recognize and apply these reflections is vital for a comprehensive understanding of function transformations.

Applying these transformations often involves a series of steps. It is crucial to understand the order in which these transformations are applied, as changing the order can lead to a different result. Typically, horizontal shifts and stretches are handled before vertical shifts and stretches, and reflections are often applied before or after shifts and stretches, depending on the context. By carefully considering the order of operations, one can accurately predict and manipulate function graphs.

Problem Statement and the Function $f(x) = 4\sqrt{x}$

Our problem involves the function $f(x) = 4\sqrt{x}$. This is a transformation of the basic square root function, $\sqrt{x}$. Let's first understand the properties of the square root function. The square root function, $y = \sqrt{x}$, is defined for non-negative values of $x$, and its graph starts at the origin (0,0) and increases as $x$ increases. The function is always non-negative, reflecting the fact that the square root of a non-negative number is non-negative. Understanding the basic shape and properties of the square root function helps in visualizing transformations applied to it.

The coefficient 4 in front of the square root represents a vertical stretch. Specifically, it means that the y-value of the transformed function will be 4 times the y-value of the original square root function for any given x. This makes the graph of $f(x)$ rise more steeply than the graph of $\sqrt{x}$. In other words, the vertical stretch by a factor of 4 makes the function's output values four times larger for the same input.

Now, let's visualize the graph of $f(x) = 4\sqrt{x}$. The graph starts at the origin (0,0), just like the basic square root function, but it rises more rapidly. For example, when $x = 1$, $f(1) = 4\sqrt{1} = 4$, whereas for the basic square root function, the y-value would be 1. Similarly, when $x = 4$, $f(4) = 4\sqrt{4} = 4 \cdot 2 = 8$, while the basic square root function would give a value of 2. This visual comparison highlights the effect of the vertical stretch.

Understanding the domain and range of $f(x)$ is also crucial. The domain of $f(x)$ is all non-negative real numbers, i.e., $x \geq 0$, because the square root is only defined for non-negative values. The range of $f(x)$ is also all non-negative real numbers, i.e., $f(x) \geq 0$, because the square root function always produces non-negative outputs, and multiplying by 4 does not change this. Knowing the domain and range helps in accurately plotting the function and understanding its behavior.

The derivative of $f(x)$ can give us additional insights into its behavior. The derivative of $f(x) = 4\sqrt{x}$ is $f'(x) = \frac{2}{\sqrt{x}}$. This shows that the slope of the tangent to the curve is always positive, indicating that the function is increasing. The derivative also tells us that the slope approaches infinity as $x$ approaches 0, reflecting the steep rise of the graph near the origin. Understanding the derivative helps in analyzing the rate of change of the function.

Applying the Transformations to Obtain $g(x)$

We are given that $g(x)$ is the graph of $f(x)$ shifted down 1 unit and right 5 units. Let's break down these transformations step by step. The first transformation is a shift down by 1 unit. This means we subtract 1 from the entire function $f(x)$. If we were to perform only this transformation, our new function would be $f(x) - 1 = 4\sqrt{x} - 1$. This vertical shift moves every point on the graph of $f(x)$ down by 1 unit, changing the y-coordinates but leaving the x-coordinates unchanged.

The second transformation is a shift right by 5 units. This means we replace $x$ with $(x - 5)$ in the function. This transformation shifts the graph horizontally along the x-axis. The horizontal shift right by 5 units affects the x-coordinates, moving each point 5 units to the right. If we apply this to the function $4\sqrt{x} - 1$, we get $4\sqrt{x - 5} - 1$.

Combining both transformations, we get the function $g(x)$. To obtain $g(x)$, we first apply the horizontal shift, replacing $x$ with $(x - 5)$ in $f(x)$, and then apply the vertical shift, subtracting 1 from the result. This gives us $g(x) = 4\sqrt{x - 5} - 1$. This is the formula for the transformed function $g(x)$, which represents the graph of $f(x)$ shifted 5 units to the right and 1 unit down.

The domain of $g(x)$ is affected by the horizontal shift. Since we are taking the square root of $(x - 5)$, we require $x - 5 \geq 0$, which means $x \geq 5$. So the domain of $g(x)$ is all real numbers greater than or equal to 5. The vertical shift affects the range of $g(x)$. The range of $f(x)$ is non-negative real numbers, but shifting down by 1 unit changes the range of $g(x)$ to all real numbers greater than or equal to -1, i.e., $g(x) \geq -1$.

Understanding the effect of these shifts on key points can further clarify the transformation. For example, in the original function $f(x)$, the point (0, 0) is the starting point. After shifting right by 5 units and down by 1 unit, this point becomes (5, -1) in the graph of $g(x)$. Similarly, if we consider the point (1, 4) on $f(x)$, which corresponds to $f(1) = 4\sqrt{1} = 4$, the shifted point on $g(x)$ would be (6, 3), as $g(6) = 4\sqrt{6 - 5} - 1 = 4\sqrt{1} - 1 = 3$. Visualizing how specific points transform helps in understanding the overall transformation of the graph.

Final Answer

Therefore, the formula for $g(x)$ is:

$g(x) = 4\sqrt{x - 5} - 1$

This represents the function $f(x) = 4\sqrt{x}$ shifted 1 unit down and 5 units to the right.