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Chapter 10: Problem 97

Given that the point (3,8) is on the graph of \(y=f(x)\) what is the corresponding point on the graph of \(y=-2 f(x+3)+5 ?\)

Short Answer

Expert verified
(0, -11)

Step by step solution

01

Substitute the Given Point

Start by noting the given point (3, 8) on the graph of the function \(y=f(x)\). Here \(x=3\) and \(f(3)=8\).
02

Horizontal Shift

The transformed function is \(y=-2 f(x+3) + 5\). For the input \(x'\) in the original function to correspond to the input \(x\) in the transformed function, set \(x+3=3\). Simplifying, \(x=0\). Thus, we will use \(f(0)\).
03

Apply the Transformation

Substitute \(x=0\) into the transformed function: \(y' = -2 f(0+3) + 5\). Since \(f(3)=8\), this becomes \(y' = -2 \times 8 + 5 = -16 + 5 = -11\).
04

Find the Corresponding Point

Our transformed input \(x' = 0\), and the corresponding output \(y' = -11\) gives the point (0, -11) on the new graph.

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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 means moving the graph left or right along the x-axis. For any function transformation of the form \( y = f(x + c) \), the graph shifts horizontally. If \( c \) is positive, the shift is to the left. If \( c \) is negative, the shift is to the right.
In our exercise, we have a transformation with \(y = -2 f(x + 3) + 5\). Here, the term \( (x + 3) \) suggests we shift the graph of \( f(x) \) to the left by 3 units. This shift changes where we evaluate our function values. Instead of evaluating at our original x-coordinate, we move left by 3. Specifically, a point originally at \( x = 3 \) becomes \( x = 3 - 3 = 0 \) in the new coordinates.
Vertical Transformation
Vertical transformations adjust a graph's position along the y-axis and affect its shape. The general form is \( y = a \, f(x) + k \), where \( a \) affects the stretch/compression (and reflection if negative), and \( k \) shifts the graph up or down.
In our transformed function \( y = -2 f(x + 3) + 5 \), the coefficient \( -2 \) reflects the graph vertically (because it's negative) and stretches it by a factor of 2. The term \( +5 \) moves the graph up by 5 units. So, any y-value of the original function is multiplied by \( -2 \), creating an inverted and stretched graph, and then increased by 5 units.
Function Graphing
Function graphing involves plotting points to create a visual representation of a function. Each point \( (x, y) \) on the graph represents an input \( x \) and its corresponding output \( y \). When functions are transformed, these points move accordingly.
To graph \( y = -2 f(x + 3) + 5 \) based on \( y = f(x) \):
  • Shift each point left by 3 units.
  • Multiply each y-value by \( -2 \), inverting and scaling the graph.
  • Move each point up by 5 units.
For example, the point \( (3, 8) \) on the original graph becomes \( (0, -11) \) after shifting horizontally, scaling/reflecting, and then applying the vertical shift.
Coordinate Transformation
Coordinate transformation refers to changing the coordinates of a point based on certain rules. It's a systematic way to find new points on a transformed graph.
Given \( y = -2 f(x + 3) + 5 \) and starting from a known point \( (3, 8) \):
1. For horizontal shift: Set \( x' = 3 \). With the transformation \( x + 3 = 3 \), solve to get \( x = 0 \).
2. Evaluate \( f \) at the adjusted x-coordinate: \( f(0 + 3) = f(3) = 8 \).
3. Apply vertical transformations: Compute \( y = -2 \times 8 + 5 = -16 + 5 = -11 \).
As a result, our new coordinates are \( (0, -11) \), showing how the original point transforms under the given function modification.

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