Problem 9 - College Algebra

Question

Write a formula for the function obtained when the graph is shifted as described. \(f(x)=\frac{1}{x^{2}}\) is shifted up 2 units and to the left 4 units.

Step-by-Step Solution

Answer

The new function is \( f(x) = \frac{1}{(x+4)^2} + 2 \).

Step 1: Understanding the Vertical Shift

To shift a function up by 2 units, we add 2 to the function’s output. For the given function, \[ f(x) = \frac{1}{x^2}, \]adding 2 gives: \[ f(x) = \frac{1}{x^2} + 2. \]

Step 2: Understanding the Horizontal Shift

To shift a function to the left by 4 units, we replace \(x\) with \(x + 4\) in the function. So replacing in \[ \frac{1}{x^2} + 2, \]we get:\[ \frac{1}{(x + 4)^2} + 2. \]

Step 3: Combining Shifts to Find the New Function

Combine the transformations from Steps 1 and 2 into a single function. The new function after shifts is:\[ f(x) = \frac{1}{(x+4)^2} + 2. \]

Key Concepts

Vertical Shift

A vertical shift in graph transformations occurs when the entire graph of a function moves up or down the y-axis. When we say that a function is shifted "up 2 units", it means we need to add 2 to the result of the function. Imagine the function's original graph simply moving upwards by 2 units without any change in its shape.

For example, consider the basic function \( f(x) = \frac{1}{x^2} \). If we apply a vertical shift upwards by 2 units, the transformed function becomes: \[ f(x) = \frac{1}{x^2} + 2 \]

The entire graph of \( \frac{1}{x^2} \) is lifted up by 2 units. This is like moving every point on the curve higher on the graph by 2 units. The vertex and the asymptotes both move vertically upward. Understanding vertical shifts is important, especially when interpreting changes to the maximum or minimum points in the graph of a function.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right along the x-axis. To achieve this, we manipulate the input value of a function. When you want to shift the graph to the left, you replace \( x \) with \( x + a \), and for shifting to the right, replace \( x \) with \( x - a \).

For our example, the function \( f(x) = \frac{1}{x^2} \) is initially horizontal-shifted to the left by 4 units. This results in replacing \( x \) with:\( x + 4 \)

Thus, the function becomes:\[ f(x) = \frac{1}{(x+4)^2} \]

This horizontal transformation means that each point on the graph moves 4 units directly to the left. The shape and orientation of the graph remain the same, but its position changes on the x-axis. Understanding the impact of horizontal shifts is crucial when analyzing where a graph intersects the y-axis or changes in patterns over the range of x values.

Function Transformations

Function transformations combine various changes to a function's graph, including vertical and horizontal shifts, reflections, stretches, and compressions. Transformations allow us to adjust the graph's appearance and position on the coordinate plane while preserving its overall form.

In the given exercise, we've combined vertical and horizontal shifts. Starting with \( f(x) = \frac{1}{x^2} \), we applied a vertical shift (up 2 units) and a horizontal shift (left 4 units). The resulting function is then transformed into:\[ f(x) = \frac{1}{(x+4)^2} + 2 \]

These transformations show us how to transform an original function into a new function that has its graph moved both vertically and horizontally. While simpler transformations affect only one aspect of the graph at a time, combined transformations can modify multiple aspects simultaneously. This understanding helps students manipulate functions graphically and algebraically, and facilitates deeper comprehension of function behavior in different contexts.