Problem 17 - Precalculus: Functions and Graphs

Question

Find all values of \(x\) such that \(f(x)>0\) and all \(x\) such that \(f(x)<0,\) and sketch the graph of \(f\) $$f(x)=x^{4}-4 x^{2}$$

Step-by-Step Solution

Answer

Positive on \((-\infty, -2)\) and \((2, \infty)\); negative on \((-2, 0)\) and \((0, 2)\).

Step 1: Analyze the Function

The function given is \(f(x) = x^4 - 4x^2\). It is a polynomial function, and our aim is to determine the intervals where the function is positive (\(f(x) > 0\)) and where it is negative (\(f(x) < 0\)). Typically, solving such inequalities involves finding roots and analyzing the sign changes around these roots.

Step 2: Find the Roots

To find where \(f(x) = 0\), solve the equation \(x^4 - 4x^2 = 0\). Factor out \(x^2\) to get \(x^2(x^2 - 4) = 0\). This gives the solutions \(x^2 = 0 \) or \(x^2 = 4\). Solving these, we find the roots \(x = 0, x = 2, x = -2\).

Step 3: Test Intervals Between Roots

The roots divide the number line into intervals: \((-\infty, -2), (-2, 0), (0, 2), (2, \infty)\). Select a test point from each interval and substitute it into \(f(x)\) to determine whether the function is positive or negative in that interval.- For \((-\infty, -2)\), test \(x = -3\): \(f(-3) = (-3)^4 - 4(-3)^2 = 81 - 36 = 45\), so \(f(x) > 0\).- For \((-2, 0)\), test \(x = -1\): \(f(-1) = (-1)^4 - 4(-1)^2 = 1 - 4 = -3\), so \(f(x) < 0\).- For \((0, 2)\), test \(x = 1\): \(f(1) = 1^4 - 4(1)^2 = 1 - 4 = -3\), so \(f(x) < 0\).- For \((2, \infty)\), test \(x = 3\): \(f(3) = 3^4 - 4(3)^2 = 81 - 36 = 45\), so \(f(x) > 0\).

Step 4: Summarize the Solution

Based on the sign tests:- \(f(x) > 0\) in the intervals \((-\infty, -2)\) and \((2, \infty)\).- \(f(x) < 0\) in the intervals \((-2, 0)\) and \((0, 2)\).

Step 5: Sketch the Graph

Begin by plotting the roots \(x = -2, x = 0, x = 2\) on the x-axis. The graph will pass through these points and change from positive to negative or vice versa as we analyzed:- The curve starts above the x-axis in \((-\infty, -2)\), drops below the x-axis between \(-2\) and \(0\), rises to pass through \(0\), drops below again until \(x = 2\), and goes above the x-axis for \(x > 2\). The end behavior, as \(x\) approaches \(\pm\infty\), moves upwards due to the \(x^4\) term.This sketch reflects the positive intervals \((-\infty, -2)\), \((2, \infty)\), and the negative intervals \((-2, 0), (0, 2)\).

Key Concepts

Roots of Polynomial Functions

Understanding the roots of a polynomial function is crucial because they are the points where the function intersects the x-axis. For the given function, \( f(x) = x^4 - 4x^2 \), finding the roots means solving \( f(x) = 0 \).
Start by factoring. The expression can be factored as \( x^2(x^2 - 4) = 0 \).

• First, solve \( x^2 = 0 \), which gives \( x = 0 \).
• Next, solve \( x^2 - 4 = 0 \), which simplifies to \( x^2 = 4 \) leading to \( x = 2 \) and \( x = -2 \).

These solutions, \( x = -2, 0, 2 \), are the roots of the polynomial. These roots are important for graph sketching and analyzing the sign of the polynomial, as they segment the number line into testable intervals.

Sign Analysis

Sign analysis helps us determine where the polynomial is positive or negative. Using our roots from the previous section, we can explore how the function behaves in different intervals. These intervals are \((-\infty, -2), (-2, 0), (0, 2), (2, \infty)\).
For each interval, choose a test point and substitute it into the polynomial to find the sign.

• For \((-\infty, -2)\), test with \(x = -3\). The result is positive, so \(f(x) > 0\).
• For \((-2, 0)\), test with \(x = -1\). The result is negative, indicating \(f(x) < 0\).
• For \((0, 2)\), test with \(x = 1\). This also produces a negative value, so \(f(x) < 0\).
• For \((2, \infty)\), test with \(x = 3\). The result is positive, which means \(f(x) > 0\).

By analyzing the sign of \(f(x)\) in each interval, we can determine where the function is above or below the x-axis.

Interval Testing

Interval testing is a systematic way to understand complex polynomial behavior by evaluating expressions at specific points.

• Each interval formed by the roots becomes a region for testing.
• In the previous section, we used random values from each of those intervals to determine if \(f(x)\) was positive or negative.

Testing intervals between roots provides insights into the function's behavior. This process finds regions where the graph is above or below the x-axis before sketching the graph.
Once the signs are established for each interval, you can predict how \(f(x)\) will behave across the entire domain, ensuring accurate representation and understanding of the polynomial's behavior.

Graph Sketching of Polynomials

Sketching the graph of a polynomial provides a visual representation of its behavior. Start by plotting the roots \(x = -2, 0, 2\) on the x-axis, because these points are where the graph crosses or touches this axis.

• Begin in the interval \((-\infty, -2)\), draw the graph above the x-axis since \(f(x) > 0\).
• At \(x = -2\), the graph will touch the axis, then dip below between \(-2\) and \(0\), since \(f(x) < 0\).
• From \(0\) to \(2\), the function remains below the x-axis.
• Past \(x = 2\), the function rises above the x-axis, producing \(f(x) > 0\).

Combine these intervals with the end behavior, noting that the graph will head upward indefinitely as \(x\) approaches \(+\infty\) and \(-\infty\). This gives a complete sketch, reflecting both the positive and negative intervals determined during interval testing.