Problem 64 - College Algebra and Calculus: An Applied Approach

Question

The point \((8,2)\) on the graph of \(f(x)=\sqrt[3]{x}\) has been shifted to the point \((5,0)\) after a rigid transformation. Identify the shift and write the new function \(h\) in terms of \(\bar{f}\)

Step-by-Step Solution

Answer

The point \((8,2)\) has been shifted 3 units to the left and 2 units down to reach the new point \((5,0)\). The new function after this rigid transformation is \(h(x) = \sqrt[3]{x+3} - 2\).

Step 1: Understanding Rigid Transformations

A rigid transformation is a transformation which preserves distances. In this problem, we have a shift of a function, which means the function has moved without changing its shape. A point \((x, y)\) on a function after a horizontal shift of \(h\) units and a vertical shift of \(k\) units takes the form \((x+h, y+k)\). The transformation from \(f(x)\) to \(h(x)\) would result in the function \(h(x) = \bar{f}(x - h) + k\.

Step 2: Identify Shifts

Look at the original point \((8,2)\) and its shifted position \((5,0)\). By comparing these two points, we can deduce that there is a horizontal shift of \(8 - 5 = 3\) units to the left, and a vertical shift of \(2 - 0 = 2\) units downwards. Hence, \(h = -3\) and \(k = -2\). The negative sign indicates the shift is to the left for \(h\) and downwards for \(k\).

Step 3: Write the New Function

Substitute the values of \(h\) and \(k\) into the transformation form of the function. The function then becomes \(h(x) = \bar{f}(x - (-3)) - 2 = \bar{f}(x + 3) - 2\). In terms of \(\bar{f}\), where \(\bar{f}(x) = \sqrt[3]{x}\) we have the new function as \(h(x) = \sqrt[3]{x+3} - 2\)

Key Concepts

Rigid Transformations

Rigid transformations are a key concept in understanding how functions can move on a graph without changing their shape or size. Imagine sliding a picture frame across a table; it stays exactly the same, just in a different location. This is what happens with a rigid transformation in math.

• Distance Preservation: In rigid transformations, the distances and angles are preserved. Therefore, any function's graph might shift position but never distorts.
• Types of Rigid Transformations: These include translations (or shifts), reflections, and rotations. However, in this problem, we're focusing purely on translations, which mean moving the graph horizontally or vertically without altering its appearance.

Learning about rigid transformations helps in visualizing how equations translate into different locations on a coordinated plane, retaining their distinct form.

Horizontal Shift

A horizontal shift in a function moves the graph left or right across the x-axis. This shift modifies the x-values of points on the graph, altering the position without changing its shape. In essence, we are adjusting where the function appears on the graph.
The rule for a horizontal shift can be understood like this:

• Shift Right: If a point \(x, y\) moves to \(x-h, y\), then the graph is shifted to the right by \(h\) units.
• Shift Left: If a point \(x, y\) moves to \(x+h, y\), then the graph is going left by \(-h\) units.

In our exercise, the graph of the function \(f(x) = \sqrt[3]{x}\) moved from \( (8, 2) \) to \( (5, 0) \), indicating a shift of 3 units to the left on the x-axis. This change is modeled mathematically by adding 3 to every x-value in the function: \(h(x) = f(x+3)\).

Vertical Shift

A vertical shift moves the graph up or down along the y-axis. This shift changes the y-values of points on the graph, again altering the position but not the shape. Understanding vertical shifts is crucial for seeing how graphs change with added or subtracted constants.
Here's how vertical shifts work:

• Shift Up: Adding a positive constant \(k\) to the function \(f(x)\) makes it \(f(x) + k\), so every point will rise \(k\) units.
• Shift Down: Subtracting a positive constant \(k\) results in \(f(x) - k\), moving every point down by \(k\) units.

In our case study, the point shifted alongside a vertical movement from \( (8, 2) \) to \( (5, 0) \), producing a shift 2 units down. This adjustment leads to modifying our function by subtracting 2 from each \(f(x)\), making way for the new equation: \(h(x) = \sqrt[3]{x+3} - 2\).