Problem 94 - Algebra and Trigonometry

Question

Graph of the Absolute Value of a Function (a) Draw the graphs of the functions \(f(x)=x^{2}+x-6\) and \(g(x)=\left|x^{2}+x-6\right| .\) How are the graphs of \(f\) and \(g\) related? (b) Draw the graphs of the functions \(f(x)=x^{4}-6 x^{2}\) and \(g(x)=\left|x^{4}-6 x^{2}\right| .\) How are the graphs of \(f\) and \(g\) related? (c) In general, if \(g(x)=|f(x)|,\) how are the graphs of \(f\) and \(g\) related? Draw graphs to illustrate your answer.

Step-by-Step Solution

Answer

The graph of \(g(x) = |f(x)|\) is \(f(x)\) with all negative y-values reflected above the x-axis.

Step 1: Understand the Function

We start by analyzing the function \(f(x) = x^2 + x - 6\). This is a quadratic function, represented in a parabola form with its vertex and intersection points being key features of its graph.

Step 2: Find Roots of the Function \(f(x)\)

To graph \(f(x)\), we first find its roots by setting \(x^2 + x - 6 = 0\). Solving gives us \(x = 2\) and \(x = -3\). These roots are where the curve intersects the x-axis.

Step 3: Find the Vertex of \(f(x)\)

The vertex of \(f(x)\) can be found using the formula \(x = -\frac{b}{2a}\), where \(a = 1\) and \(b = 1\). This yields \(x = -0.5\). Plugging this back into \(f(x)\), we get the vertex point \((-0.5, -6.25)\).

Step 4: Graph Function \(f(x)\) and \(g(x) = |x^2 + x - 6|\)

Plot the parabola using the roots and vertex. For \(g(x)\), reflect any part of the graph of \(f(x)\) that is below the x-axis above it, as the absolute value makes all y-values non-negative.

Step 5: Analyze \(f(x) = x^4 - 6x^2\)

This function is a quartic (4th degree) polynomial that can lead to a w-shaped curve. Factor to get roots: \(x^2(x^2 - 6) = 0\). This gives \(x = 0\) and \(x = \pm\sqrt{6}\).

Step 6: Graph \(f(x) = x^4 - 6x^2\) and \(g(x) = |x^4 - 6x^2|\)

Plot based on the roots and the fact it is symmetrical with respect to the y-axis. Reflect any downward parts above the x-axis to graph \(g(x)\).

Step 7: General Relationship between \(f(x)\) and \(g(x) = |f(x)|\)

Graph of \(g(x)\) is obtained by reflecting sections of \(f(x)\) that fall below the x-axis across it. The parts of \(f(x)\) above the x-axis remain unchanged in \(g(x)\).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra, and they are typically represented by the formula \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a parabola, which can either open upwards or downwards depending on the sign of \( a \). If \( a > 0 \), the parabola opens upwards, resembling a U-shape. Conversely, if \( a < 0 \), it opens downwards like an inverted U.
When graphing a quadratic function, it's crucial to identify the vertex, axis of symmetry, and roots (or x-intercepts). The vertex, the highest or lowest point on the graph, can be found using the formula \( x = -\frac{b}{2a} \), and you substitute this back into \( f(x) \) to find the y-coordinate. The roots are obtained by solving the equation \( ax^2 + bx + c = 0 \).
In our problem, the quadratic function is \( f(x) = x^2 + x - 6 \). By solving \( x^2 + x - 6 = 0 \), we find the roots \( x = 2 \) and \( x = -3 \). The vertex can be calculated as \( x = -\frac{1}{2} \), making it \( (-0.5, -6.25) \). With this information, we can sketch the parabola, showing the direction it opens, its roots, and vertex.

Quartic Functions

Quartic functions are polynomials of the fourth degree. They can take the form \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \), with \( a eq 0 \). The graph of a quartic function can display a variety of shapes, including a wavy, W-shaped or M-shaped curve, depending on the leading coefficient's sign and the number of turning points.
When considering the function \( f(x) = x^4 - 6x^2 \), we are dealing with a simpler quartic function, as it lacks third-degree and linear terms. This particular quartic function is symmetric about the y-axis. To find the roots, you can set \( x^4 - 6x^2 = 0 \) and factor it as \( x^2(x^2 - 6) = 0 \). Solving gives roots at \( x = 0 \) and \( x = \pm \sqrt{6} \).
The function opens upwards because the leading term \( x^4 \) has a positive coefficient. The positive sign causes the ends of the graph to rise to infinity as \( x \) moves away from the origin. This understanding, along with finding the intercepts and any potential local maxima or minima, is crucial for sketching an accurate graph of a quartic function like this one.

Graph Transformations

Graph transformations are methods used to manipulate the graph of a function, to create a new graph. A common transformation involves the absolute value function, which changes negative y-values to positive. When dealing with \( g(x) = |f(x)| \), any section of the graph of \( f(x) \) that lies below the x-axis is reflected above it. Thus, the parts of \( f(x) \) that are already positive remain unchanged.
Let's consider how this applies to the functions mentioned in the exercise:

• For \( f(x) = x^2 + x - 6 \), the negative y-values below the x-axis become positive in \( g(x) \), creating a "V" shape at the vertex point, where the parabola dips below the x-axis before reflection.
• For \( f(x) = x^4 - 6x^2 \), the downward portions of the W-shaped graph are reflected above the x-axis, making \( g(x) \) appear mirror-like symmetrical about the x-axis.

Understanding these transformations is key in modifying functions for various applications, and this simple reflection technique is often used in graphing absolute value functions.