Problem 32 - College Algebra

Question

Tabular representations for the functions \(f, g,\) and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\). $$ \begin{array}{|c|r|r|r|r|r|} \hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & -1 & -3 & 4 & 2 & 1 \\ \hline \end{array} $$ $$ \begin{array}{|c|r|r|r|r|r|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & -1 & -3 & 4 & 2 & 1 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{h}(\boldsymbol{x}) & -2 & -4 & 3 & 1 & 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

\(g(x) = f(x+1)\), \(h(x) = f(x) - 1\).

Step 1: Examine g(x) relative to f(x)

The function values for \(g(x)\) are \([-1, -3, 4, 2, 1]\) associated with the x-values \([-3, -2, -1, 0, 1]\). Compare this to \(f(x)\), which has the same function values \([-1, -3, 4, 2, 1]\) for x-values \([-2, -1, 0, 1, 2]\). This indicates that \(g(x)\) is a horizontal shift of \(f(x)\) by 1 unit to the left.

Step 2: Write transformation for g(x)

Since \(g(x)\) matches the values of \(f(x)\) but at x-values shifted one unit left, the transformation is \(g(x) = f(x+1)\).

Step 3: Examine h(x) relative to f(x)

Function \(h(x)\) has the values \([-2, -4, 3, 1, 0]\) at the same x-values \([-2, -1, 0, 1, 2]\) as \(f(x) = [-1, -3, 4, 2, 1]\). Notice that each value of \(h(x)\) is 1 less than the corresponding \(f(x)\), indicating a vertical shift down by 1 unit.

Step 4: Write transformation for h(x)

Since \(h(x)\) is one unit less than \(f(x)\) for each corresponding x-value, the transformation is \(h(x) = f(x) - 1\).

Key Concepts

Horizontal Shift

Function transformations can be thought of as changes to the position, shape, or size of a graph. One such transformation is the horizontal shift, which involves moving the graph of a function left or right. To understand this better, let's delve into the concept.

A horizontal shift occurs when each point on the graph of a function is moved horizontally by the same distance. For example, if we have a function \( f(x) \) and it is transformed into \( g(x) = f(x + 1) \), this indicates a shift to the left by 1 unit.

When comparing the functions \( f(x) \) and \( g(x) \), notice that the y-values remain constant, but the x-values for \( g(x) \) are consistently smaller by 1 than those of \( f(x) \). This confirms a leftward move. Thus, the operation inside the function \( f(x) \) manipulates the input to achieve this transformation.

• "\( + \)" in the argument of a function represents a shift to the left.
• "\( - \)" would represent a shift to the right.

The idea of horizontal shifts is crucial as it helps us predict and adjust functions without recalculating all values. It enhances understanding and visualization of algebraic functions.

Vertical Shift

Vertical shifts are another fundamental aspect of function transformations. Unlike horizontal shifts, which involve the x-axis, vertical shifts affect the y-axis. To illustrate, consider the transformations of the function \( f(x) \).

The transformation of \( h(x) = f(x) - 1 \) is an example of a vertical shift. Each value of \( h(x) \) in our example is exactly 1 unit less than the corresponding value of \( f(x) \). This means the entire graph of \( f(x) \) has moved downward by 1 unit.

• When a constant is subtracted from \( f(x) \), the graph shifts downward.
• Conversely, if a constant was added, the graph would move upwards.

This transformation keeps the x-values unchanged, ensuring that the shift is only vertical. Such shifts are useful in graphing specific ranges of functions or adjusting the output values without altering the input values.

Remember, vertical shifts are applied directly to the function's output, making them straightforward.

Tabular Representation

Tabular representation is a powerful tool for understanding function transformations. It uses tables to list different x-values and their corresponding y-values, helping to visually identify shifts and other transformations.

In our exercise, tables are used to compare \( f(x), g(x), \) and \( h(x) \). By closely observing these tables:

• We see that \( g(x) \) maintains the same y-values as \( f(x) \) but has shifted x-values, indicating a horizontal shift.
• Meanwhile, \( h(x) \) shares the same x-values with \( f(x) \) but has uniformly decreased y-values, showing a vertical shift.

Using tables can be incredibly helpful to track transformations systematically. They provide clear, organized data, making patterns more apparent.

By practicing with tabular representation, students can gain a thorough grasp of how transformations affect both the x and y-values in functions. Understanding these changes not only aids in graphing functions but also in comprehending their behavior across different representations.