Question

Let \(f(x)=x^3-4 x^2+6\). Write a rule for \(g\). Describe the graph of \(g\) as a transformation of the graph of \(\boldsymbol{f}\). $$ g(x)=\frac{1}{2} f(x)-3 $$

Short answer

The function \(g(x)\) is a scaled (by a factor of \(\frac{1}{2}\)) and vertically translated (downwards by 3 units) version of the function \(f(x)\). The rule for \(g(x)\) can be written as \(g(x) = \frac{1}{2}x^3 - 2x^2\).

Step-by-step solution

Identify the Transformation Operations

Look at the equation \(g(x) = \frac{1}{2}f(x) - 3\). It can be seen that there are two operations being performed on the function \(f(x)\). The first operation is scaling the function by a factor of \(\frac{1}{2}\), this means the output of the function \(f(x)\) is halved. The second operation is a vertical shift or translation of the function by -3 units. This means the output of the function \(f(x)\) is shifted downward by 3 units.

Write the Rule for \(g(x)\)

Substitute the function \(f(x)=x^3-4 x^2+6\) into the equation for \(g\). Thus, the rule for \(g(x)\) is given by:\(g(x) = \frac{1}{2}(x^3 - 4x^2 + 6) - 3\).

Simplify the Rule for \(g(x)\)

Distribute the \(\frac{1}{2}\) factor and combine constants to simplify the rule:\(g(x) = \frac{1}{2}x^3 - 2x^2 + 3 - 3\).This simplifies to:\(g(x) = \frac{1}{2}x^3 - 2x^2 + 0\).

Describe the Transformation

This function \(g(x)\) is the graph of the function \(f(x)\) which has been scaled by a factor of \(\frac{1}{2}\) and then vertically shifted downwards by 3 units.

Key concepts

Function Transformations

Function transformations involve changing the graph of a base function into a new graph—typically by stretching, compressing, shifting, or reflecting.

Understanding how to transform functions is essential when analyzing the behavior of equations. In algebra, function transformations can be categorized into several types, including scaling, translating (shifting), and reflecting. To visualize a transformation, imagine manipulating the shape of a graph in various ways without altering its original input-output relationship.

For instance, when looking at the equation given as an exercise, \(g(x) = \frac{1}{2}f(x) - 3\), we see that two operations are performed on the function \(f(x)\). Recognizing these operations is the first step in understanding function transformations, allowing us to write a new rule for the transformed function \(g(x)\) as illustrated.

Scaling Functions

Scaling a function changes its size without altering its shape. It's akin to either stretching or compressing the graph of the function along the y-axis (vertical scaling) or the x-axis (horizontal scaling).

In our exercise, the scaling factor of \( \frac{1}{2} \) indicates a vertical compression by 50%. It's like taking the original graph and squeezing it towards the x-axis, which makes every y-value half of what it was on the original graph of \( f(x) \).

To better understand this, let's apply the \( \frac{1}{2} \) factor to the function \(f(x)\) as specified in the rule for \(g(x)\), effectively reducing the function's maximum and minimum points by half and hence transforming the original graph into a vertically scaled version.

Vertical Shifts

A vertical shift is when we move the entire graph of a function up or down without changing its shape or horizontal position. This movement is determined by a constant added or subtracted from the function's rule.

In the context of our exercise, subtracting 3 from the function \( \frac{1}{2}f(x)\) results in shifting the graph downwards by 3 units. It's like taking the already scaled graph and moving it lower on the y-axis, affecting each point on the graph equally. This is particularly useful when trying to align graphs with new horizontal or vertical aspects, such as axes or other lines.

Combining the concepts of scaling and shifting is essential for mastering graph transformations. By applying these two transformations to \(f(x)\), we obtain the final transformed function \(g(x)\), showing the power of algebra in modifying and understanding complex functions.