Problem 53 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-\frac{1}{2}\left(x^{2}-4\right)\left(x^{2}-1\right)$$

Step-by-Step Solution

Answer

End behavior: \(y=-\frac{1}{2}x^{4}\). The x-intercepts are at -2, -1, 1, 2, and y-intercept is at 0. The function is positive when -2 < x < -1 or 1 < x < 2 and The function is negative when -infinity < x < -2, -1 < x < 1, 2 < x < infinity. The graph should reflect all these points and behaviors.

Step 1: Determine the End Behavior

A function of the form \(y=c x^{2}\) represents a simple parabola. The end behavior of the parabola and the given function would be the same if the highest degree term of the polynomial is considered. The highest power in the given polynomial is 4, which is on terms \(x^{2}\) in both bracket. Also, a multiplication of these will give \(x^{4}\). So, copy this along with coefficient -1/2 and ignore all terms of lower degree. Therefore, the function becomes \(y=-\frac{1}{2}x^{4}\).

Step 2: Find the x and y intercepts

The x-intercept of the polynomial is the value of x that makes the polynomial equal to zero. From \(f(x)=0\), we get \(-\frac{1}{2}\left(x^{2}-4\right)\left(x^{2}-1\right)=0\). Solving this gives x = -2, -1, 1, 2. The y-intercept is the value of the polynomial at x = 0. So, by substituting \(x=0\) in \(f(x)\), we get \(f(0)=0\). Therefore, the x-intercepts are -2, -1, 1, 2 and y-intercept is 0.

Step 3: Find the intervals where the function is positive

To find when the function is positive, consider each of the expressions \(x^{2}-4\) and \(x^{2}-1\). These are positive when \(-2 < x < -1\) or \(1 < x < 2\). This will yield \(f(x)= -\frac{1}{2}\left(x^{2}-4\right)\left(x^{2}-1\right)\) being positive.

Step 4: Find the intervals where the function is negative

To find when the function is negative, consider each of the expressions \(x^{2}-4\) and \(x^{2}-1\). These are negative when \(-\infty< x< -2\) or \(-1< x < 1\) or \(2< x <\infty\). This will yield \(f(x)= -\frac{1}{2}\left(x^{2}-4\right)\left(x^{2}-1\right)\) being negative.

Step 5: Sketch the Graph

Using all the information from steps 1 to 4, a graph can be drawn. The x-intercepts are at -2, -1, 1, 2. The y-intercept is at 0. The function is positive between -2 and -1, and between 1 and 2. The function is negative between -infinity and -2, between -1 and 1, and between 2 and infinity. The sketch should reflect all these characteristics, and the ends of the graph should approach positive and negative infinity as x approaches negative and positive infinity, respectively, keeping in mind the nature of the function \(y=-\frac{1}{2}x^{4}\).

Key Concepts

End Behavior

End behavior describes how a polynomial function behaves as the input, or x-values, approach infinity or negative infinity. For polynomial functions, the end behavior is strongly influenced by the highest degree term. In the given polynomial, the highest degree term is influenced by both \(x^2 - 4\) and \(x^2 - 1\). When these are expanded and multiplied, the term with the highest degree is \(x^4\). So, focusing on this highest power, we represent the end behavior using \(y = -\frac{1}{2}x^4\). This emphasizes that the function will approach negative infinity as \(x\) becomes very large or very negative because the coefficient \(-\frac{1}{2}\) is negative. This tells us the arms of the polynomial graph fall down on both sides.

X-Intercepts

The x-intercepts of a polynomial function are the points where the graph crosses the x-axis. This happens when the function equals zero. For the polynomial \( f(x) = -\frac{1}{2}(x^2 - 4)(x^2 - 1) \), set \(f(x) = 0\). This gives the equation \( (x^2 - 4)(x^2 - 1) = 0 \), which factors further to find the roots \( x = -2, -1, 1, \text{ and } 2 \). Therefore, these x-values are the x-intercepts of the graph. At these points, the value of the polynomial function is zero, and the graph touches or crosses the x-axis.

Y-Intercepts

To find the y-intercepts of a polynomial function, we set \(x\) to zero and determine what \(f(x)\) becomes. For our polynomial \( f(x) = -\frac{1}{2}(x^2 - 4)(x^2 - 1) \), substituting \(x = 0\) results in \( f(0) = -\frac{1}{2}((-4)(-1)) = 0 \). This indicates that the y-intercept occurs at the origin, \( (0, 0) \), meaning the graph passes through this point. Y-intercepts are helpful in understanding the starting point of the function when graphing.

Positive and Negative Intervals

The positive and negative intervals of a polynomial function describe the sections of the graph where the function exists above or below the x-axis, respectively. For our polynomial, positive intervals occur where \(f(x)\) is above the x-axis. After solving \( f(x) = -\frac{1}{2}((x^2 - 4)(x^2 - 1)) > 0 \) for positive values, we find \(-2 < x < -1\) and \(1 < x < 2\) are the intervals where the function is positive.
Negative intervals are where the function dips below the x-axis. Solving \( f(x) < 0 \) reveals negative intervals occurring at \(-\infty < x < -2\), \(-1 < x < 1\), and \(2 < x < \infty\). Understanding these intervals is crucial for knowing how the graph fluctuates.

Graph Sketching

Graph sketching combines all previous findings—end behavior, x and y intercepts, and positive and negative intervals—to create a visual representation of the polynomial function. Start by plotting the x-intercepts at \(-2, -1, 1,\) and \(2\), and the y-intercept at the origin \((0, 0)\).
The end behavior shows both sides of the graph will descend visibly downward as \(x\) approaches extreme values from either direction, reflecting the negative leading coefficient of the highest degree term \(-\frac{1}{2}x^4\).
Next, highlight where the curve lies above or below the x-axis, using the positive (\(-2 < x < -1\) and \(1 < x < 2\)) and negative (\(-\infty < x < -2\), \(-1 < x < 1\), \(2 < x < \infty\)) intervals. This method leads to a sketch which accurately portrays all characteristics of the polynomial, offering a solid visual understanding of its behavior.