Question

Determine the function that is graphed if the graph of \(f(x)=\sqrt{x}\) is reflected about the \(x\) -axis and then vertically compressed by a factor of \(\frac{1}{3}\)

Short answer

\( g(x) = -\frac{1}{3} \sqrt{x} \)

Step-by-step solution

Identify the Original Function

The original function given is \( f(x) = \sqrt{x} \).

Reflect About the x-axis

Reflecting the function about the \(x\)-axis means multiplying the function by -1. Thus, the reflected function becomes \( -\sqrt{x} \).

Apply Vertical Compression

After reflecting, compress the function vertically by multiplying it by \( \frac{1}{3} \). Hence, the vertically compressed function is \( -\frac{1}{3} \sqrt{x} \).

Write the Final Function

The final function, after reflecting about the \(x\)-axis and then compressing vertically by a factor of \( \frac{1}{3} \), is \( g(x) = -\frac{1}{3} \sqrt{x} \).

Key concepts

Reflection About x-axis

When a function is reflected about the x-axis, it essentially flips over the x-axis. This type of transformation takes every point of the original function and moves it to the opposite side of the x-axis. The y-values of each point become their negatives.

For example, given the function \(f(x) = \sqrt{x} \), reflecting it about the x-axis gives us \(-f(x) = -\sqrt{x} \).

Visualizing this on a graph, every point \((a, b)\) on \(f(x)\) becomes \((a, -b)\) on \(-f(x)\).

This change mirrors the graph, flipping it upside down while maintaining the same x-values.

Vertical Compression

A vertical compression pushes the graph towards the x-axis, making it 'flatter'. To compress a function vertically by a factor of \(k\), you multiply the function by \(k\), where \(0 < k < 1\).

In our example, after reflecting \(f(x) = \sqrt{x} \) to get \( -\sqrt{x}\), we compress it by a factor of \( 1/3 \). This gives us:

\[ g(x) = -\frac{1}{3}\sqrt{x} \].

The effect of this transformation is that the y-values of the original function are one third of their previous values, making the graph less steep.

Square Root Function

The square root function, \( f(x) = \sqrt{x} \), is a fundamental function in mathematics.
It produces the non-negative square root of x.

• The domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \), as you cannot take the square root of a negative number and get a real number.
• The range of \( f(x) = \sqrt{x} \) is also \( y \geq 0 \) because square roots produce non-negative results.

Graphically, \( f(x) = \sqrt{x} \) starts at the origin (0,0) and increases slowly, curving upwards.
As x increases, \( f(x) \) increases but at a decreasing rate.

Understanding the behavior of the square root function is crucial for applying transformations like reflection and vertical compression.