Problem 66 - Calculus Early Transcendentals

Question

Describe how the graph of \(g\) is obtained from the graph of \(f .\) (a) \(g(x, y)=f(x-2, y) \quad\) (b) \(g(x, y)=f(x, y+2)\) (c)\(g(x, y)=f(x+3, y-4)\)

Step-by-Step Solution

Answer

(a) Right by 2 units; (b) Down by 2 units; (c) Left by 3 units, up by 4 units.

Step 1: Understand the Function Transformations

When transforming functions, we adjust their position on the graph. For a function \(g(x, y)\) obtained from \(f(x, y)\), we pay attention to shifts or translations. For an expression \(f(x-h, y-k)\), the graph of \(f\) moves horizontally by \(h\) units and vertically by \(k\) units.

Step 2: Apply Horizontal Translation for (a)

In part (a), we have \(g(x, y) = f(x-2, y)\). This expression indicates a horizontal shift. Specifically, it means the graph of \(f\) is shifted to the right by 2 units because we subtract 2 from \(x\).

Step 3: Apply Vertical Translation for (b)

In part (b), we have \(g(x, y) = f(x, y+2)\). This indicates a vertical shift. The graph of \(f\) is shifted down by 2 units, as adding 2 to \(y\) corresponds to moving downward.

Step 4: Apply Both Horizontal and Vertical Translations for (c)

In part (c), \(g(x, y) = f(x+3, y-4)\). This involves both horizontal and vertical shifts. The graph of \(f\) shifts left by 3 units due to \(+3\) added to \(x\) and up by 4 units due to \(-4\) added to \(y\).

Key Concepts

Horizontal Shift

In the world of functions, a horizontal shift is a transformation that moves the graph of a function left or right. This is typically represented in the formula as modifying the value of the input variable, usually the 'x' value. For instance, consider the function transformation described by \( g(x, y) = f(x-2, y) \). This tells us that the graph of the original function \( f \) has been shifted horizontally.
When you see \( f(x-h, y) \), it indicates a shift to the right by 'h' units if \( h \) is positive, and to the left if \( h \) is negative. So, subtracting 2 from \( x \) as in our example \( g(x, y) = f(x-2, y) \) means that every point on the graph of \( f \) is moved to the right by 2 units.
- Moves graph left: Add to \( x \) - Moves graph right: Subtract from \( x \)
This transformation does not change the shape or orientation of the graph, only its position along the x-axis.

Vertical Shift

A vertical shift involves moving the graph of a function up or down along the y-axis. This is done by altering the output variable, affecting the 'y' value. As seen in part (b) of the original problem, the transformation \( g(x, y) = f(x, y+2) \) involves such a vertical shift.
In a vertical shift, adding a value to \( y \), like \( f(x, y+k) \), shifts the graph down by 'k' units. Conversely, subtracting 'k' units shifts it up. In our case, adding 2 to \( y \) means moving the graph down by 2 units.

• Moves graph up: Subtract from \( y \)
• Moves graph down: Add to \( y \)

This adjustment affects the vertical position without altering the graph's size, shape, or horizontal location.

Graph Translation

Graph translation combines both horizontal and vertical shifts to position a graph at any desired location on a coordinate plane. It effectively allows for moving a graph along both axes simultaneously, preserving its overall shape and orientation.
The function transformation \( g(x, y) = f(x+3, y-4) \) illustrated in part (c) involves both types of shifts. It reveals the graph of \( f \) is translated left by adding 3 to \( x \) and upward by subtracting 4 from \( y \).

• The shift left is caused by \( x+3 \), indicating a move by 3 units.
• The shift up results from \( y-4 \), representing a move upwards by 4 units.

These translations do not alter the fundamental shape or size of the graph, only relocating it to a new position. This method is particularly useful when trying to align graphs in comparison or when adjusting for specific coordinate constraints.