Question

Consider the function \(f(x)=\frac{1}{4} \sqrt{5 x}\). a) Identify the transformations represented by \(f(x)\) as compared to \(y=\sqrt{x}\). b) Write two functions equivalent to \(f(x)\): one of the form \(y=a \sqrt{x}\) and the other of the form \(y=\sqrt{b x}\). c) Identify the transformation(s) represented by each function you wrote in part b). d) Use transformations to graph all three functions. How do the graphs compare?

Short answer

The function \(f(x)\) is a combination of vertical and horizontal compressions. Equivalent forms are \(y=\frac{\sqrt{5}}{4} \sqrt{x}\) and \(y=\sqrt{\frac{5}{16}x}\). Graphs show the compressions visually.

Step-by-step solution

- Identify Transformations in f(x)

Examine the given function, which is \(f(x)=\frac{1}{4} \sqrt{5x}\). This can be broken down into two transformations from the base function \(y=\sqrt{x}\): a vertical compression by a factor of \(\frac{1}{4}\) and a horizontal compression by a factor of \(5\).

- Write Function in the Form y=a\sqrt{x}

To rewrite \(f(x)\) in the form \(y=a \sqrt{x}\), combine the constants into a single multiplier: \(f(x)=\frac{1}{4}\sqrt{5} \sqrt{x}\). Therefore, the equivalent function is \(y=\frac{\sqrt{5}}{4} \sqrt{x}\).

- Write Function in the Form y=\sqrt{bx}

To rewrite \(f(x)\) in the form \(y=\sqrt{bx}\), factor out the constants under a single square root: \(f(x)=\frac{1}{4} \sqrt{5x} = \sqrt{\frac{5x}{16}}\). Thus, the equivalent function is \(y=\sqrt{\frac{5}{16}x}\).

- Identify Transformations in Equivalent Forms

For \(y=\frac{\sqrt{5}}{4} \sqrt{x}\), there is a vertical compression of \(\frac{\sqrt{5}}{4}\). For \(y=\sqrt{\frac{5}{16} x}\), there is a horizontal compression of \(\frac{5}{16}\).

- Graphing the Functions

Graph \(y=\sqrt{x}\), \(y=\frac{\sqrt{5}}{4}\sqrt{x}\), and \(y=\sqrt{\frac{5}{16} x}\). The first graph is the base function. \(y=\frac{\sqrt{5}}{4} \sqrt{x}\) will be vertically compressed compared to the base function and \(y=\frac{\sqrt{5}}{4} \sqrt{x}\) will be horizontally compressed. Compare their shapes and identify any shifts or stretches.

Key concepts

Graph Transformations

When we talk about graph transformations, we are referring to changes that alter the appearance of a graph. Transformations include shifts, stretches, compressions, and reflections. Each type of transformation adjusts the graph in a different way.

For the given function, the base function is the square root function, which is written as: \( y = \sqrt{x} \)

The transformations applied to this particular function are vertical and horizontal compressions. These types of transformations are crucial for understanding how the graph will differ from the original. By examining transformed graphs, you can predict the new shape and position of a function's graph.

Here's a simple summary of the types of graph transformations:

• **Translations:** Shifting the graph up, down, left, or right.
• **Reflections:** Flipping the graph over a specific axis (x-axis or y-axis).
• **Stretches and Compressions:** Scaling the graph either vertically or horizontally.

Vertical Compression

Vertical compression involves squishing the graph towards the x-axis. This transformation occurs when the function includes a coefficient that is a fraction (between 0 and 1) multiplied by the original function.

For the given function, we observe a vertical compression by a factor of \( \frac{1}{4} \): \( f(x)=\frac{1}{4} \sqrt{5x} \)

This means the function's output values are scaled down. For every original y-value of the base function, the new y-value will only be a quarter as much. If the base function output is y, the compressed output will be \( \frac{y}{4} \).

• **Example Calculation:** If \( y=4 \) in the original function, in the compressed function, it will be \( \frac{4}{4} = 1 \).

Understanding vertical compression helps to predict how the graph's height will change. Lower peaks and shallower valleys characterize a vertically compressed graph.

Horizontal Compression

Horizontal compression squeezes the graph towards the y-axis. This occurs when a coefficient multiplies the variable inside the function itself, rather than outside.

For our function, the horizontal compression factor is 5: \( f(x)=\stop{1}{4} \sqrt{5x} \)

This compression impacts the input values instead of the output. Here’s how it reflects in the function's behavior: For every value of x in the original function, the new x-value will be compressed by a factor of 5. This means if the base function crosses x at 1, the compressed function will cross at 0.2.

• **Example Calculation:** If \( x=5 \) in the base function, the new x-value will be \( \frac{5}{5} = 1 \).

Horizontal compression results in the graph appearing narrower. This squeezing effect drastically changes how the function looks when plotted.

Equivalent Functions

Equivalent functions are different expressions of the same function that yield the same outputs for corresponding inputs. They might look different algebraically, but they represent the same underlying relationship.

Let's convert our given function to two equivalent forms:

• **First Form (y=a \sqrt{x}):** Simplify constants: \( f(x)=\frac{1}{4} \sqrt{5} \sqrt{x} \) becomes \( y=\frac{\sqrt{5}}{4} \sqrt{x} \)
  
• **Second Form (y=\sqrt{bx}):** Combine constants under the root: \( f(x)=\frac{1}{4} \sqrt{5x} = \sqrt{\frac{5x}{16}} \) becomes \( y=\sqrt{\frac{5}{16}x} \)
  

Each form contains information about vertical and horizontal compressions. In the first equivalent form, you see a clear vertical compression by \( \frac{\sqrt{5}}{4} \), while in the second form, a horizontal compression by \( \frac{5}{16} \) is evident. Both forms reflect equivalent graphical transformations, ensuring the function's behavior remains consistent.