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Chapter 2: Problem 30

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 \(\frac{3}{5}\) unit

Short Answer

Expert verified
The final equation is \(y = \frac{1}{2}\sqrt[3]{-x} + \frac{3}{5}\).

Step by step solution

01

Reflect across the y-axis

When you reflect a function across the y-axis, you replace every instance of \(x\) in the function with \(-x\). The function \(f(x) = \sqrt[3]{x}\) becomes \(f(x) = \sqrt[3]{-x}\).
02

Vertical Shrink by Factor of 1/2

To apply a vertical shrink by a factor of \(\frac{1}{2}\), you multiply the entire function by \(\frac{1}{2}\). So, \(\sqrt[3]{-x}\) becomes \(\frac{1}{2}\sqrt[3]{-x}\).
03

Shift Upward by 3/5 Unit

To shift the graph upward by \(\frac{3}{5}\) units, you add \(\frac{3}{5}\) to the function. Therefore, \(\frac{1}{2}\sqrt[3]{-x}\) becomes \(\frac{1}{2}\sqrt[3]{-x} + \frac{3}{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection
Reflection is a fascinating way to transform a function's graph. When we talk about reflecting over the y-axis, we're essentially flipping the graph horizontally.
To achieve this, we replace every instance of the variable \( x \) in the function with \( -x \). Let's take the function \( f(x) = \sqrt[3]{x} \) for example.
Reflecting this function over the y-axis gives us \( f(x) = \sqrt[3]{-x} \).
  • Reflection over the y-axis: Change \( x \) to \( -x \).
  • The graph becomes a mirror image across the y-axis.
When you reflect a graph, it doesn't alter the shape of the curve or the distance between points. We're just flipping it over like turning a page in a book.
Vertical Shrink
A vertical shrink is all about compressing the graph toward the x-axis. This transformation affects how tall or short the function looks, making it narrower vertically.
To apply this shrink, you multiply the entire function by some factor. In our exercise, the factor is \( \frac{1}{2} \). For the function \( f(x) = \sqrt[3]{-x} \), a vertical shrink by \( \frac{1}{2} \) means you get \( f(x) = \frac{1}{2}\sqrt[3]{-x} \).
  • Vertical shrink compresses the graph downwards.
  • Multiply the function by the shrink factor \( \frac{1}{2} \).
This transformation reduces the height of every point on the graph, but the x-coordinates remain the same. It's like gently squeezing the graph towards the axis underneath.
Vertical Shift
The vertical shift is a simple yet effective transformation to move a graph up or down. To do this, you either add or subtract a constant from the entire function.
In our case, we need to move the graph of \( \frac{1}{2}\sqrt[3]{-x} \) up by \( \frac{3}{5} \) units.We simply add \( \frac{3}{5} \) to the whole function to elevate every point on the graph. The final function becomes: \( f(x) = \frac{1}{2}\sqrt[3]{-x} + \frac{3}{5} \).
  • Shift upwards by adding \( \frac{3}{5} \) to the function.
  • This moves the entire graph higher on the grid.
Vertical shifts do not change the shape of the graph; they simply move the entire curve vertically without affecting its horizontal position.

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