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Chapter 1: Problem 42

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

Short Answer

Expert verified
\(f(-x)\) is a reflection of \(f(x)\) across the y-axis.

Step by step solution

01

Understand the Original Function

The original function is given as \( f(x) \). This function has a graph that depends on the specific rule or form of \( f(x) \). Our task is to understand how the transformation \( f(-x) \) changes the graph of this function.
02

Identify the Transformation Type

When we replace \( x \) with \( -x \) in the function, the transformation applied to the original function \( f(x) \) is a reflection across the y-axis. This means each point \((x, y)\) on the graph of \( f(x) \) is moved to \((-x, y)\) for \( f(-x) \).
03

Describe the Impact on the Graph

Reflecting across the y-axis changes the direction of the graph horizontally but keeps the vertical positions along the y-axis the same. This does not affect the graph's y-intercepts or general height of features but changes the orientation of horizontal features.

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Key Concepts

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

Reflection Across the Y-Axis
Reflection across the y-axis is a fascinating transformation that alters the graph of a function. When a function, like \(f(x)\), undergoes this transformation, it becomes \(f(-x)\), which means its graph is mirrored across the y-axis. To visualize this, imagine folding the graph along the y-axis. Every point on the right is swapped with its counterpart on the left, and vice versa.
  • This transformation does not change the heights of points (y-values).
  • It only affects the horizontal position, flipping points over the y-axis.
  • The curve keeps its original shape, but its direction across the x-axis is reversed.
Understanding this transformation helps in analyzing how the function's behavior changes visually and behaviorally, helping many concepts click into place.
Original Function
The original function, \(f(x)\), is the starting point for any transformation. It's crucial to know its basic properties and graph before we apply transformations like reflection. To delve deeper into this topic:
  • The original function's form determines its graph鈥攍inear, quadratic, or otherwise.
  • Understanding subtle details like where the key points and intercepts lie is important.
  • These details will contribute to anticipating what transformed versions of the function might look like.
Moreover, by analyzing \(f(x)\), you create a foundation. This foundation makes it easier to predict and understand how transformations, like reflections, alter its representation.
Graph of Function
The graph of a function is a powerful visual tool that illustrates how a function behaves. It serves as a map, showing every point that makes the equation true. When you apply transformations, like a reflection across the y-axis, it changes the graph correspondingly.
  • Reflections change the graph's orientation but not the vertical arrangement.
  • Key features like intercepts along the y-axis remain the same.
  • The overall height and steepness are unchanged during a y-axis reflection.

Thus, when focusing on how \(f(x)\) becomes \(f(-x)\), it鈥檚 helpful to visualize these horizontal flips and recognize the symmetry this creates. This perspective provides a simple way to foresee the function's changes and reinforces the importance of visualizing mathematical concepts.

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Most popular questions from this chapter

Determine the interval(s) on which the function is concave up and concave down. $$ m(x)=-2(x+3)^{3}+1 $$

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

Describe how each function is a transformation of the original function \(f(x)\). $$ 3 f(-x) $$

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

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