Question

\(f(x)=x^3-6\); translation 3 units left, followed by a reflection in the \(y\)-axis

Short answer

The new function is \(f(-x + 3) = (-x + 3)^3 - 6\).

Step-by-step solution

Translation

The first transformation is a horizontal translation (shift) of the function 3 units to the left. This is achieved by replacing the variable x in the function with \(x+3\). Thus, the translated function is \(f(x + 3) = (x + 3)^3 - 6\).

Reflection

The second transformation is a reflection about the y-axis. This is achieved by replacing the variable x in the translated function with \(-x\). Therefore, the reflected function is \(f(-x + 3) = (-x + 3)^3 - 6\).

Final Result

Putting together the transformations, the function \(f(x) = x^3 - 6\), after being shifted 3 units to the left and reflected about the y-axis, is transformed into the new function \(f(-x + 3) = (-x + 3)^3 - 6\).

Key concepts

Horizontal Translation

Understanding horizontal translation is pivotal when dealing with function transformations. It involves shifting the graph of a function left or right along the x-axis. When a function is translated horizontally, every point on the graph moves the same distance in the same direction. For example, to translate the function \(f(x) = x^3-6\) three units to the left, you would replace every x in the function with (x + 3). This gives you the new function \(f(x + 3) = (x + 3)^3 - 6\). The addition of 3 to x is counterintuitive; although we are moving left, we add because the function's graph moves in the opposite direction of what happens to x.

To visualize horizontal translations, imagine sliding your function on a number line: a positive change slides to the right, while a negative change slides it to the left. This understanding is vital for mastering more complex transformations in algebra.

Reflection about the y-axis

Reflection about the y-axis is another intriguing transformation. It flips the graph of a function across the y-axis, resulting in a mirror image. To reflect a function about the y-axis, replace x with -x in the function's rule. For the horizontally translated function \(f(x + 3) = (x + 3)^3 - 6\), reflecting it about the y-axis changes x to -x, resulting in \(f(-x + 3) = (-x + 3)^3 - 6\).

Visualizing this, if the original function has a point (a, b), after reflection, this point will be located at (-a, b). It is crucial to comprehend that the function's shape remains the same, but its orientation is inverted.

Algebraic Manipulation

Mastering algebraic manipulation is essential when working with function transformations. It involves rewriting expressions and functions in different forms to facilitate operations such as translations and reflections. In the context of the prior transformations, algebraic manipulation is used to apply the necessary changes to the variable x. The process might involve expanding powers, simplifying expressions, or factoring.

For instance, to complete the transformation of our function, one could expand \((-x + 3)^3\) to see the detailed changes applied to each term. Understanding algebraic manipulation can help in identifying the effects of each transformation step on the original function and is a cornerstone for tackling higher-level mathematical problems.

Function Notation

Function notation is a method of writing algebraic relationships in a standardized form, which clearly indicates operations to be performed on variables. For our function \(f(x) = x^3 - 6\), f represents the function itself, while (x) shows that x is the independent variable. After applying transformations, we see the notation \(f(-x + 3) = (-x + 3)^3 - 6\), signaling that the function should be evaluated with -x + 3 as the input.

Nailing down function notation is immensely important for understanding and communicating mathematics effectively. It helps in determining what a function looks like after transformations and serves as a blueprint for understanding and constructing further mathematical concepts.