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Chapter 3: Problem 30

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-|x| $$

Short Answer

Expert verified
The graph of \(f(x) = -|x|\) is a downward-opening V-shape with its vertex at the origin.

Step by step solution

01

Identify the Standard Function

The given function is \(f(x) = -|x|\). The standard function associated with this is the absolute value function \(g(x) = |x|\). The graph of \(g(x) = |x|\) is a V-shaped graph with its vertex at the origin (0,0), opening upwards.
02

Determine the Transformation

The transformation present in \(f(x) = -|x|\) is a reflection. Specifically, the negative sign in front of the absolute value indicates a reflection over the x-axis. This means wherever the standard function \(g(x) = |x|\) has values, they will be inverted across the x-axis in \(f(x) = -|x|\).
03

Sketch the Graph

Begin by sketching the standard V-shaped graph \(g(x) = |x|\) with the vertex at the origin. Now, reflect this graph over the x-axis to get the graph of \(f(x) = -|x|\). The resulting graph will still have its vertex at (0,0) but will open downwards.

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Key Concepts

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

Absolute Value Function
The absolute value function is a fundamental concept in algebra. This function is denoted as \(|x|\), and it represents the distance of a number \(x\) from zero on the number line. Because distance cannot be negative, the absolute value is always zero or a positive number.

When we graph the absolute value function, its most notable feature is its symmetry and V-shape. The basic form \(y = |x|\) consists of two linear segments meeting at a point called the vertex, which is at the origin \((0,0)\).
  • On the left side, the graph declines as \(x\) becomes more negative.
  • On the right side, the graph inclines as \(x\) becomes more positive.
  • The vertex is a point of symmetry where the two sides of the V merge.
The absolute value function is used in various mathematical situations, such as calculating deviations and solving equations that involve distances.
Reflection Over x-axis
Reflection over the x-axis is a transformation that flips the graph of a function upside down. In mathematical terms, when you reflect a function \(y = f(x)\) over the x-axis, you change it to \(y = -f(x)\).

For the absolute value function \(y = |x|\), this reflection results in a new function \(y = -|x|\). Each point on the graph of \(y = |x|\) moves directly downward, (inverting across the x-axis), resulting in the graph flipping its direction:
  • The vertex stays the same, at the origin \((0,0)\).
  • The previously upward opening V-shape now opens downward.
This transformation is visually intuitive as it appears like holding a paper with the graph and flipping it over a horizontal line, demonstrating how negative values impact graph orientation.
V-shaped Graph
A V-shaped graph typically refers to the graph of an absolute value function. It's distinctive due to its sharp vertex and linearly extending branches on both sides.

For \(y = |x|\), its V-shaped graph is centered at the vertex \((0, 0)\), creating a very recognizable shape.
  • The point at the bottom or top (depending on transformation) of the V is the vertex.
  • The arms of the V are perfectly linear, neither curved nor bent.
  • This V-shape is useful for identifying key transformation effects, such as translations and reflections.
When the graph is transformed, such as reflecting over the x-axis to \(y = -|x|\), the pointy bottom becomes a peak, as the graph opens downwards while retaining its original V-shape characteristics.

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