Problem 30 - Algebra and Trigonometry

Question

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt[3]{x}\) , reflect in the \(y\) -axis, shrink vertically by a factor of \(\frac{1}{2},\) and shift upward \(\frac{1}{3}\) unit

Step-by-Step Solution

Answer

\(f(x) = \frac{1}{2}\sqrt[3]{-x} + \frac{1}{3}\) is the final transformed function.

Step 1: Reflect in the y-axis

To reflect the graph of a function in the y-axis, we replace \( x\) with \(-x\). Therefore, the function \( f(x) = \sqrt[3]{x} \) becomes \( f(x) = \sqrt[3]{-x} \). This transformation reflects the graph across the y-axis.

Step 2: Shrink Vertically by a Factor of 1/2

A vertical shrink by a factor of \( \frac{1}{2} \) is achieved by multiplying the entire function by \( \frac{1}{2} \). So, the function becomes \( f(x) = \frac{1}{2}\sqrt[3]{-x} \). This reduces the height of the graph by half its original size.

Step 3: Shift Upward by 1/3 Unit

To shift the graph upward by \( \frac{1}{3} \) unit, we add \( \frac{1}{3} \) to the function. Therefore, the transformed function is \( f(x) = \frac{1}{2}\sqrt[3]{-x} + \frac{1}{3} \). This step results in the final position of the graph after all transformations.

Key Concepts

Reflecting Functions

Reflecting a function across the y-axis is a type of transformation that involves flipping the graph over the y-axis. This is specifically done by transforming the independent variable in the function. If you have a function defined as \(f(x)\), reflecting it in the y-axis requires replacing \(x\) with \(-x\). For example, if we apply this to the function \(f(x) = \sqrt[3]{x}\), the transformed function becomes \(f(x) = \sqrt[3]{-x}\). By doing this, every point \((x, y)\) on the original graph is mapped to \((-x, y)\) on the transformed graph.

In simpler terms, reflecting across the y-axis can be visualized as if the graph stands in front of a mirror along the y-axis. This is a handy technique when you need to analyze how a function behaves symmetrically in reverse.

Vertical Shrink

A vertical shrink in function transformations refers to compressing the graph of the function towards the x-axis. This is carried out by multiplying the entire function by a number between 0 and 1—known as the "shrink factor." Suppose we have \(f(x) = \sqrt[3]{-x}\), and we intend to perform a vertical shrink by a factor of \(\frac{1}{2}\). In this case, we would multiply the function by \(\frac{1}{2}\), resulting in the new function \(f(x) = \frac{1}{2}\sqrt[3]{-x}\).

Think of it like squeezing the graph from top to bottom, making it vertically shorter. Each point \((x, y)\) on the original graph is moved to \((x, \frac{y}{2})\), effectively halving the height of the graph's output values. This transformation does not affect the x-values or the horizontal spacing between points, only the vertical stretch of the graph.

Vertical Shift

Vertical shifting is a straightforward transformation where you slide the graph of a function up or down. To achieve a vertical shift, you simply add or subtract a constant from the entire function. If we consider the function \(g(x) = \frac{1}{2}\sqrt[3]{-x}\) and we want to shift it upward by \(\frac{1}{3}\) unit, we add \(\frac{1}{3}\) to the function: resulting in \(f(x) = \frac{1}{2}\sqrt[3]{-x} + \frac{1}{3}\).

This means that every point on the graph of the function is moved directly upward by that amount. Using this shift technique, if a point on the unshifted graph was at \((x, y)\), it will now be at \((x, y+\frac{1}{3})\). Such translations are crucial for analyzing the effect of adding constant values to functions, often used to correct or shift the baseline of a graph in data representation.