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Chapter 1: Problem 7

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x)-2 $$

Short Answer

Expert verified
The function \( f(x) - 2 \) is a vertical shift downward by 2 units.

Step by step solution

01

- Identify the Original Function

The original function in this context is given as \( f(x) \). This function serves as the base or starting point for any transformations that we apply.
02

- Understand Function Notation

The expression \( f(x) - 2 \) signifies that we are modifying the output of the function \( f(x) \) by subtracting 2 from it. This modification represents a vertical transformation.
03

- Recognize the Type of Transformation

Subtracting a constant from a function, as in \( f(x) - 2 \), corresponds to a vertical shift. Specifically, the entire graph of the function \( f(x) \) is shifted downward by 2 units.
04

- Graphical Interpretation

To visualize this transformation, imagine the graph of \( f(x) \). Each point on this graph will move 2 units downwards. This does not affect the shape of the graph, only its vertical position on the coordinate plane.

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Key Concepts

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

Vertical Shift
A vertical shift in a function occurs when a constant number is added to or subtracted from the function. It changes the entire graph's position along the y-axis. For the problem at hand, the transformation is expressed as \( f(x) - 2 \). This signifies a vertical shift downwards by 2 units for every output value of the function \( f(x) \).
  • When you subtract a number, the graph moves downward.
  • When you add a number, it shifts upwards.
Importantly, a vertical shift does not change the graph's shape or alter any of its other properties, such as slope or curvature. It simply repositions the graph, maintaining all other aspects of its structure.
Graph Interpretation
Graph interpretation helps you see the effects of function transformations visually. In the case of the function transformation \( f(x) - 2 \), imagine plotting the original function \( f(x) \) on a graph. Each point of the \( f(x) \) graph moves exactly 2 units downward to create the graph of \( f(x) - 2 \).
  • The points' paths are parallel and maintain the same distance from each other.
  • The x-coordinates remain unchanged.
This kind of transformation allows you to identify how new graphs derive from the original function visually. Despite the shift, the graph's general shape, such as its peaks, valleys, or linearity, does not alter.
Function Notation
Function notation is a way to name functions and understand transformations applied to them. It uses a combination of letters and often mirrors mathematical operations. Here, \( f(x) \) represents the output of a function given an input \( x \). The notation \( f(x) - 2 \) clearly shows that for each output of the function \( f(x) \), we subtract 2.
  • This format makes it easy to understand what transformations are being applied.
  • The notation helps maintain clarity as functions grow more complex with more operations.
Through function notation, one can easily grasp complex alterations to functions, aiding in problem-solving and graph analysis.

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Most popular questions from this chapter

Sketch a graph of each function as a transformation of a toolkit function. $$ k(x)=(x-2)^{3}-1 $$

Suppose \(f(x)=x^{2}+8 x-4\). Compute the following: $$ \text { a. } f(-1)+f(1) \quad \text { b. } f(-1)-f(1) $$

Sketch a graph of each function as a transformation of a toolkit function. $$ f(t)=(t+1)^{2}-3 $$

For each of the following functions, evaluate: \(f(-2), f(-1), f(0), f(1),\) and \(f(2)\). $$ f(x)=\frac{x-3}{x+1} $$

Tables of values for \(f(x), g(x),\) and \(h(x)\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\). $$ \begin{aligned} &\begin{array}{|c|c|} \hline x & f(x) \\ \hline-2 & -1 \\ \hline-1 & -3 \\ \hline 0 & 4 \\ \hline 1 & 2 \\ \hline 2 & 1 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline x & g(x) \\ \hline-3 & -1 \\ \hline-2 & -3 \\ \hline-1 & 4 \\ \hline 0 & 2 \\ \hline 1 & 1 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline x & h(x) \\ \hline-2 & -2 \\ \hline-1 & -4 \\ \hline 0 & 3 \\ \hline 1 & 1 \\ \hline 2 & 0 \\ \hline \end{array} \end{aligned} $$
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