Problem 18 - Algebra and Trigonometry

Question

15–18 An equation and its graph are given. Find an inequality whose solution is the shaded region. $$y=x^{3}-4 x$$

Step-by-Step Solution

Answer

The inequality is \( x^3 - 4x \geq 0 \).

Step 1: Understand the Graph and Equation

The graph given is of the function \( y = x^3 - 4x \). This is a cubic function and its graph is defined by the points where the function equal zero, as well as the end behavior of the cubic function.

Step 2: Determine the Zeros of the Function

Find the zeros of the equation \( y = x^3 - 4x \). Set the equation equal to zero: \( x^3 - 4x = 0 \). Factor the equation as \( x(x^2 - 4) = 0 \). From this, determine \( x = 0 \), \( x = 2 \), and \( x = -2 \) as the zeros.

Step 3: Analyze the Function’s Behavior

Evaluate how the function behaves around its zeros. For \( x < -2 \), the function is negative. Between \( x = -2 \) and \( x = 0 \), it's positive. Between \( x = 0 \) and \( x = 2 \), it's negative again, and for \( x > 2 \), the function is positive.

Step 4: Identify the Shaded Region

The shaded region on the graph corresponds to the solutions where the function is either greater than or less than zero. Assuming the shaded region involves \( y \geq 0 \), we need to find the inequality that corresponds to these sections of the graph.

Step 5: Construct the Inequality

Since the shaded region corresponds to \( y \geq 0 \), and our function describes \( y \), we can write the inequality as \( x^3 - 4x \geq 0 \). This inequality includes the sections where our function is above the x-axis and equal to the zeros.

Key Concepts

Cubic Functions

A cubic function is a polynomial function of degree three, which means its highest degree of the term is three. These functions can be expressed in the general form: \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants and \( a \) is not equal to zero.

Cubic functions have:

• Up to three real zeros or roots.
• An "S" shaped curve in their graph.
• One point of inflection where the curvature changes direction.

One important characteristic of cubic functions is their end behavior. As \( x \to \infty \), the function tends to \( + o \infty \) if the leading coefficient is positive, and \( - o \infty \) if it's negative. Similarly, as \( x \to - o \infty \), it goes toward the opposite of these values. These behaviors help to understand and sketch the graph of cubic functions systematically.

Graph Interpretation

Interpreting graphs of functions involves understanding not only the basic shape but also key features such as intercepts, slopes, minimums and maximums, and other significant points.

For cubic functions like \( y = x^3 - 4x \), you can use the following observations:

• The graph crosses the x-axis at its zeros, which in this case are \( x = -2 \), \( x = 0 \), and \( x = 2 \).
• Between these points, the function alternates between positive and negative values, which tells you which portions of the graph are above or below the x-axis.
• The point of inflection typically occurs at the midpoint of the x-values of the zero crossings, causing the curve to change shape slightly.

Interpreting shaded regions in a graph helps determine where a function is positive or negative along its domain, which can be crucial when solving inequalities. Recognizing these regions allows you to interpret inequalities represented in the graph as all x-values where the graph is shaded."},{

Zeros of Functions

Zeros of a function, often referred to as roots or solutions, are the values of \( x \) that make the function equal to zero. These are the points where the graph of the function crosses the x-axis.

To find the zeros, set the function equal to zero and solve for \( x \). For \( y = x^3 - 4x \), this involves:

• Setting \( x^3 - 4x = 0 \), which can be factored as \( x(x^2 - 4) = 0 \).
• Further factoring \( x^2 - 4 \) as \( (x - 2)(x + 2) \), giving the solutions \( x = 0 \), \( x = 2 \), and \( x = -2 \).

These zeros are critical for understanding the behavior of the function. Knowing where the zeros are helps determine the intervals where the function is positive or negative as it informs how the curve moves around these points. Additionally, zeros are essential for solving inequalities, since they define the boundaries of regions where the function’s value either satisfy or do not satisfy an inequality condition.