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Chapter 2: Problem 30

If the point \(P\) is on the graph of a function \(f\) find the corresponding point on the graph of the given function. $$P(-2,4) ; \quad y=\frac{1}{2} f(x-3)+3$$

Short Answer

Expert verified
The corresponding point is \( (1, 5) \).

Step by step solution

01

Identify the Original Point

The point given is \( P(-2, 4) \). This means that when \( x = -2 \), \( f(x) = 4 \).
02

Apply the Horizontal Shift

The function \( y = \frac{1}{2} f(x-3) + 3 \) includes a horizontal shift. The term \( x - 3 \) indicates that there is a shift to the right by 3 units. Therefore, adjust the original \( x \)-value: \( x = -2 + 3 = 1 \).
03

Apply the Vertical Stretch and Shift

The function involves a vertical stretch by a factor of \( \frac{1}{2} \) and a vertical shift up by 3 units. Start by multiplying the original \( y \)-value by \( \frac{1}{2} \): \( \frac{1}{2} \times 4 = 2 \). Then, apply the vertical shift by adding 3: \( 2 + 3 = 5 \).
04

Determine the Corresponding Point

After applying the transformations, the new point \( P' \) on the transformed graph is at \( (1, 5) \).

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Key Concepts

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

Horizontal Shift
A horizontal shift in a function entails moving the graph left or right along the x-axis. This occurs without altering the shape of the function. When you see a function with a modification like \( f(x - c) \), it means each point on the graph is moved right by \( c \) units.
In our exercise, the function \( y = \frac{1}{2} f(x-3) + 3 \) includes \( x - 3 \), which shifts the graph to the right by 3 units.
  • Original point was at \( x = -2 \).
  • New position after horizontal shift: \( x = -2 + 3 = 1 \).
Shifting horizontally is a simple translation of the x-coordinate, maintaining the y-value initially. Hence, the horizontal shift does not alter the y-value at this step, as demonstrated in the procedure above.
Vertical Stretch
A vertical stretch impacts the steepness or height of the graph along the y-axis. It is achieved by multiplying the y-values of the function by a constant factor.For the function in the example, \( y = \frac{1}{2}f(x-3) + 3 \), there is a vertical stretch factor of \( \frac{1}{2} \).
This modifies each y-value by multiplying it by \( \frac{1}{2} \):
  • Original y-value: \( 4 \).
  • Stretched y-value: \( \frac{1}{2} \times 4 = 2 \).
This vertical stretch compresses the graph because the factor \( \frac{1}{2} \) is less than 1, reducing the y-value's magnitude. It's vital to consider how vertical stretches impact the graph's overall appearance, making it "shorter" or "taller" based on factors greater or less than 1.
Vertical Shift
A vertical shift involves moving the entire graph up or down along the y-axis.
This transformation adds or subtracts a constant factor from the y-values of the function.In our case, \( y = \frac{1}{2} f(x-3) + 3 \) specifies a vertical shift up by 3 units. Here's how it works:
  • Y-value after vertical stretch: \( 2 \).
  • Adjusted y-value after vertical shift: \( 2 + 3 = 5 \).
This step moves the graph upwards, raising every point by 3 units on the y-axis. Vertical shifts are crucial for aligning graphs precisely along the y-axis when visualizing and comparing different functions.
Coordinate Transformation
Coordinate transformation encompasses the collective changes that a function undergoes such as shifts, stretches, and compressions. It's about viewing the entire graph transformation as a result of each individual adjustment. Collectively, these shifts and stretches determine the location of each new point.For our exercise example, we initially had:
  • Original point: \( P(-2, 4) \).
After applying:
  • A horizontal shift (3 units to the right): \( x = 1 \).
  • A vertical stretch (factor of \( \frac{1}{2} \)): new y-value \( 2 \).
  • A vertical shift (up 3 units): final y-value \( 5 \).
All these actions yield the transformed point \( P'(1, 5) \). Understanding the complete coordinate transformation helps connect the effects of each individual shift and stretch, resulting in the new configuration of the graph.

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