Problem 30 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(f(x) = -|x|\).

Step 3: 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.

Key Concepts

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.