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Chapter 13: Problem 67

In each part, describe in words how the graph of \(g\) is related to the graph of \(f .\) $$\begin{array}{ll}{\text { (a) } g(x, y)=f(x-1, y)} & {\text { (b) } g(x, y)=1+f(x, y)} \\ {\text { (c) } g(x, y)=-f(x, y+1)} & {}\end{array}$$

Short Answer

Expert verified
(a) Shift right by 1; (b) Shift up by 1; (c) Shift down by 1 and reflect across x-axis.

Step by step solution

01

Identify the type of transformation for function (a)

For part (a), the function is given by \( g(x, y) = f(x-1, y) \). This indicates a horizontal translation. The function \( f(x, y) \) is moved 1 unit to the right. The subtraction in the \(x\)-coordinate (\(x-1\)) suggests this direction.
02

Determine transformation for function (b)

In part (b), the function is \( g(x, y) = 1 + f(x, y) \). This adds a constant to the entire function, representing a vertical shift. The graph of \( f(x, y) \) is shifted 1 unit upwards.
03

Analyze transformation for function (c)

Part (c) has the function \( g(x, y) = -f(x, y+1) \). This involves two transformations. First, \( f(x, y+1) \) suggests a translation 1 unit downwards in the \(y\)-direction. Secondly, the multiplication by \(-1\) reflects the graph across the \(x\)-axis, flipping it upside down.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Translation
Horizontal translation involves shifting a function along the horizontal axis. In our exercise, we see this with the function \( g(x, y) = f(x-1, y) \). Here, the original function \( f(x, y) \) undergoes a shift to the right by 1 unit.
This happens because we replace \( x \) with \( x - 1 \). The subtraction sign indicates the direction of the shift.
  • If we subtract a number from \( x \) (like \( x - 1 \)), the graph moves to the right that many units.
  • If we add a number (like \( x + 1 \)), the graph moves to the left.
This transformation leaves the graph's shape unchanged, only altering its position along the \( x \)-axis.
Vertical Translation
Vertical translation means moving the graph up or down along the vertical axis. For function \( g(x, y) = 1 + f(x, y) \), a constant is added to the entire function.
This specific transformation shifts the graph of \( f(x, y) \) upwards by 1 unit.
  • When a constant is added to the function (\(+1\)), it moves the graph up.
  • If a constant were subtracted (like \(-1\)), the graph would shift down.
This type of transformation doesn't change the shape or orientation of the graph; it only moves the entire graph up or down.
Reflection Across Axis
Reflecting a graph involves flipping it over a given axis. In our example, \( g(x, y) = -f(x, y+1) \), there's a combination of transformations.
First, \( f(x, y+1) \) translates the graph downwards by 1 unit. But crucially, the negative sign \(-f(x, y+1)\) reflects the graph across the \( x \)-axis.
  • Flipping across the \( x \)-axis reverses the graph's direction, turning it upside down.
  • A similar reflection across the \( y \)-axis would mirror the graph from left to right.
This transformation changes the graph's orientation, giving it an upside-down appearance when reflected over the \( x \)-axis.

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