Question

Let \(f(x)=\sqrt[3]{x}\). Write a rule for \(g\) that represents the indicated transformation of the graph of \(f\). \(g(x)=f(-x)+3\)

Short answer

The rule for the transformed function \(g(x)\) is \(g(x) = \sqrt[3]{-x} + 3\).

Step-by-step solution

Identify the transformations

The expression \(g(x)=f(-x)+3\) implies two transformations of the function \(f(x) = \sqrt[3]{x}\)1. The function \(f(-x)\) reflects the function \(f(x)\) on the y-axis.2. The function \(f(x) + 3\) shifts the function \(f(x)\) upwards by 3

Apply the transformations

The first transformation \(f(-x) = \sqrt[3]{-x}\) which reflects the graph of \(f(x) = \sqrt[3]{x}\) on the y-axis.The second transformation \(f(x) + 3 = \sqrt[3]{-x} + 3\) shifts the graph of \(f(x) = \sqrt[3]{-x}\) upwards by 3

Key concepts

Graph Reflection

Graph reflection is a fascinating transformation concept that essentially "flips" a graph over a specified axis. In the context of our function, we're focusing on reflection over the y-axis.

This means that every point on the graph of the original function is mirrored across the vertical axis. For the cubic root function specified as \( f(x) = \sqrt[3]{x} \), reflecting this graph over the y-axis involves changing the sign of the input value \( x \). As a result, our new function becomes \( f(-x) = \sqrt[3]{-x} \).

What happens here is quite intuitive. Each point \((x, y)\) on the original graph is transformed to \((-x, y)\) on the reflected graph. So, a point at \((2, 1)\) would move to \((-2, 1)\). This reflection essentially reverses the direction horizontally, creating a mirror image across the y-axis.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down in the Cartesian plane. In our problem, we're dealing with a shift in the positive y-direction, which means moving upward.

To accomplish this, we add a constant to the output value of the function. Given our reflected function, \( f(x) = \sqrt[3]{-x} \), adding 3 creates the transformation \( g(x) = \sqrt[3]{-x} + 3 \). This operation shifts every point on the graph upwards by 3 units.

Imagine taking the entire graph and nudging it straight up so that what was at \( y = 0 \) is now at \( y = 3 \), \( y = 1 \) becomes \( y = 4 \), and so forth. This is a useful transformation when you're adjusting models to align with a shift in a baseline constant or when considering changes in elevation on a graph.

Cubic Root Function

The cubic root function is a type of radical function that can express both positive and negative numbers, giving it a unique "S" shape graph spanning all quadrants. The standard form is \( f(x) = \sqrt[3]{x} \), and it represents the principal root of \( x \).

This function is distinct because its domain includes all real numbers, unlike the square root function constrained to non-negative numbers. Its graph passes through the origin \((0, 0)\) and features reflectional symmetry across the origin.

The transformations discussed, such as reflections and vertical shifts, can significantly alter the appearance and position of the cubic root graph. By applying these adjustments, one manipulates the basic shape and slope, useful for solving various algebraic and practical applications that harness the properties of roots.