Problem 85 - Algebra and Trigonometry

Question

\(85-86\) . These exercises show how the graph of \(y=|f(x)|\) is obtained from the graph of \(y=f(x)\) . The graphs of \(f(x)=x^{2}-4\) and \(g(x)=\left|x^{2}-4\right|\) are shown. Explain how the graph of \(g\) is obtained from the graph of \(f .\)

Step-by-Step Solution

Answer

Reflect parts of \( f(x) \) below the x-axis above it to obtain \( g(x). \)

Step 1: Understanding the Function f(x)

The function \( f(x) = x^2 - 4 \) is a quadratic function. Its graph is a parabola opening upward with its vertex at (0, -4). This parabola intersects the x-axis at \( x = -2 \) and \( x = 2 \) which are its roots, as \( x^2 - 4 = (x-2)(x+2) \).

Step 2: Analyze the Absolute Value Function g(x)

The function \( g(x) = |f(x)| = |x^2 - 4| \) involves taking the absolute value of \( f(x) \). This means that any part of the graph of \( f(x) \) that is below the x-axis will be reflected above the x-axis, because the absolute value function takes all negative outputs and makes them positive.

Step 3: Identify Changes in the Graph

For \( f(x) = x^2 - 4 \), the parts of the graph that are below the x-axis occur between \( x = -2 \) and \( x = 2 \). In \( g(x) \), the section of the graph between \( x = -2 \) and \( x = 2 \) will be reflected so that "U" shaped portion of the original graph below the x-axis becomes a "∩" like shape above the x-axis, making all \( f(x) \) values positive.

Step 4: Plot the Transformed Graph

To sketch \( g(x) \), begin by plotting \( f(x) = x^2 - 4 \) as usual. Then, for values of \( x \) between \(-2\) and \(2\), reflect the segment below the x-axis to be above the x-axis. Validate by checking symmetry along the y-axis and noting the graph stays unchanged for \( x < -2 \) and \( x > 2 \) because these sections are non-negative in \( f(x) \).

Key Concepts

Absolute Value Function

The absolute value function is a unique mathematical operation that transforms every number into its non-negative form. It is commonly denoted as \( |x| \), which translates any negative input to a positive and leaves positive inputs unchanged.
For example, \( |3| = 3 \) and \( |-3| = 3 \). This makes the absolute value function particularly useful when dealing with graphs, as it impacts the portions that fall below the x-axis.

When applying the absolute value function to another function, such as \( g(x) = |f(x)| \), any negative y-values (outputs) of \( f(x) \) are reflected above the x-axis. This is evident in the transformation of the graph of \( f(x) = x^2 - 4 \) into \( g(x) = |x^2 - 4| \).
In this scenario, the segment of the parabola below the x-axis between \( x = -2 \) and \( x = 2 \) is reflected upwards, transforming an original downward opening "U" shape into an upward "∩" shape at the bottom of the graph.

Quadratic Functions

Quadratic functions are polynomial functions of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. They are characterized by their U-shaped graphs called parabolas.
One key feature of quadratic functions is their symmetry. The graph of a quadratic function has a vertical line of symmetry known as the axis of symmetry, which divides the parabola into two mirror-image halves. The point at which this axis intersects the parabola is called the vertex.

• For \( f(x) = x^2 - 4 \), the vertex is at the point \((0, -4)\).
• This particular quadratic function opens upwards (since the coefficient of \( x^2 \) is positive) and has an axis of symmetry along the y-axis (\( x = 0 \)).

The quadratic function intersects the x-axis at its roots, which can be found by solving \( ax^2 + bx + c = 0 \). For \( f(x) = x^2 - 4 \), the roots are \( x = -2 \) and \( x = 2 \).
The behavior of the graph changes dramatically when the absolute value operator is applied, as parts below the x-axis are altered.

Parabola

A parabola is the graphical representation of a quadratic function. It is a symmetrical curve where each point is equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
In the context of the function \( f(x) = x^2 - 4 \), the parabola opens upwards because the coefficient of \( x^2 \) is positive. Its vertex, the lowest point on the graph, is located at \((0, -4)\).

• Key features of a parabola include its vertex, axis of symmetry, and direction of opening.
• The x-intercepts (roots) further define its shape, and for \( f(x) = x^2 - 4 \), these occur at \( x = -2 \) and \( x = 2 \).

When transforming this parabola through the absolute value function to form \( g(x) = |x^2 - 4| \), the graph undergoes a significant transformation.
The downward section of the parabola (between \( x = -2 \) and \( x = 2 \)) is mirrored above the x-axis, resembling an upside-down parabola at the bottom of the graph. This creates a new graph with two "arms" pointing upwards.