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Chapter 1: Problem 23

Make a conjecture about the relationship between the graph of \(y=f(x)\) and the graph of \(y=|f(x)|\); check your conjecture with some specific functions.

Short Answer

Expert verified
\(|f(x)|\) reflects negative parts of \(f(x)\) above the x-axis.

Step by step solution

01

Understanding the Function Transformations

When dealing with transformations like moving from \(y = f(x)\) to \(y = |f(x)|\), the effect is that all the negative values of \(f(x)\) get reflected above the x-axis. This means the parts of the graph that were below the x-axis reflect to become positive, while the points at or above the x-axis remain unchanged.
02

Analyze Specific Function 鈥� Linear Function

Consider the function \(f(x) = x - 2\). The graph of \(y = f(x)\) crosses the x-axis at \(x = 2\). The part of the line below the x-axis (for \(x < 2\)) flips above it in \(y = |f(x)|\). Therefore, for \(x < 2\), the graph of \(y = |f(x)|\) will have the equation \(y = -(x - 2) = 2 - x\).
03

Analyze Specific Function 鈥� Quadratic Function

Consider the function \(f(x) = x^2 - 4\), a parabola that opens upwards with roots at \(x = -2\) and \(x = 2\). Below the x-axis for \(-2 < x < 2\), the graph is reflected above the x-axis for \(y = |f(x)|\). The section of \(f(x)\) between the roots becomes positive, effectively making the graph of \(y = |f(x)|\) a 'W' shape with the bottom part of \(y = f(x)\) flipped.
04

Observe General Pattern

In general, when moving from \(y = f(x)\) to \(y = |f(x)|\), any part where \(f(x)\) is negative is reflected vertically above the x-axis, while parts where \(f(x)\) is positive remain unchanged. This occurs always, resulting in a graph that looks like the original for positive sections and mirrored for negatives.

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Key Concepts

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

Function Transformations
When you transform a function, you convert its graph or shape in a specific way, often to make it easier to understand or analyze. One common transformation is changing the graph of a function like \( y = f(x) \) to \( y = |f(x)| \).
This particular transformation affects only the portions of the graph that are negative.
Here's what happens:
  • The negative segments of the function are reflected upwards.
  • All the values at or above the x-axis stay exactly as they are.
This transformation is crucial because it changes the entire feel of the graph without affecting the x-values directly. Beyond absolute value, other transformations might involve stretching, compressing, or translating a graph. But with absolute values, the key move is flipping those negative pieces up to the positive side.
Graph Reflections
Graph reflections can be a fascinating aspect of visualizing mathematical functions. When reflecting the graph of a function \( y = f(x) \) about the x-axis using absolute values, you create \( y = |f(x)| \).
This process affects the negative portions beneath the x-axis, flipping them upward.
Some points to consider:
  • If \( f(x) \) is negative at a point, \( |f(x)| \) will mirror it above the x-axis.
  • If \( f(x) \) is zero or positive, it remains unchanged.
Such reflections are a subset of transformations that help retain the positive nature of the graph. They can provide insights into the symmetry and behavior of functions, offering a mirrored view of negative values. This technique is especially helpful in analyzing functions like quadratics and linear equations, where different parts of the curve behave divergently.
Linear Functions
Linear functions represent the simplest type of functions that form straight lines when graphed. The general form of a linear function is \( f(x) = ax + b \).
Analyzing their behavior with absolute value transformations can provide valuable insights.
Consider the example \( f(x) = x - 2 \):
  • The graph crosses the x-axis at \( x = 2 \).
  • For \( x < 2 \), the function yields negative values.
  • When you apply \( y = |f(x)| \), those negative sections are mirrored upwards, creating a V-like shape.
This transformation preserves the intersection point at \( x = 2 \) while ensuring all graphed values are non-negative, fitting neatly into a real-world context where an outcome must be above a baseline, like measuring time or distance.
Quadratic Functions
Quadratic functions are illustrated by parabolas and have the general form \( f(x) = ax^2 + bx + c \).
Their graphs are distinct because of their U-shape or inverted U-shape, depending on the coefficient of the \( x^2 \) term.
Consider \( f(x) = x^2 - 4 \):
  • This quadratic has roots at \( x = -2 \) and \( x=2 \).
  • Between these roots, the graph dips below the x-axis.
  • With \( y = |f(x)| \), this section flips above, leading to a W-shaped graph.
Such transformations alter the graph's appearance significantly yet maintain key characteristics like vertex and axis of symmetry. By flipping just the negative portion, we see how mathematical behaviors adjust while retaining core features, making quadratic analysis with absolute values a dynamic and insightful exercise.

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Most popular questions from this chapter

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility A vertical line intersecting the \(x\) -axis at \(x=2,\) oriented upward.

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