Problem 33 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$g(x)=-2(x+1)^{2}(x-3)^{2}$$

Step-by-Step Solution

Answer

The end behavior of the function is that as \(x\) approaches both positive and negative infinity, \(g(x)\) approaches negative infinity. The \(y\)-intercept is at (0,18). The \(x\)-intercepts are at -1 and 3 with multiplicity 2. The function presents no symmetry. The function is positive or negative at intervals determined by the x-intercepts -1 and 3.

Step 1: Identify the End Behavior

End behavior of a function can be predicted from the degree and leading coefficient of the function. For even degree and negative leading coefficient as in our function \(g(x)=-2(x+1)^{2}(x-3)^{2}\), both ends of the function will point downwards. So as \(x\) approaches positive infinity, \(g(x)\) approaches negative infinity, and as \(x\) approaches negative infinity, \(g(x)\) also approaches negative infinity.

Step 2: Finding the y-intercept

The y-intercept is the function value when \(x = 0\). To find this, plug \(x = 0\) into the function to get \(g(0) = -2(0+1)^{2}(0-3)^{2} = -2(1)(-3)^2 = 18 . The y-intercept is at (0, 18).

Step 3: Finding the x-intercepts and multiplicities of the real zeros

The x-intercepts are found when the \(y = 0\). In the function, setting \(g(x) = 0\) gives -2(x+1)^{2}(x-3)^{2} = 0. This implies that \(x=-1\) and \(x=3\) are the zeros or x-intercepts. Both \(x = -1\) and \(x = 3\) have multiplicity 2 since the factors (x+1) and (x-3) are raised to the power of 2.

Step 4: Finding the symmetries of the graph of the function

A function has symmetry if it is either even or odd. An even function has symmetry about the y-axis, while an odd function has symmetry about the origin. To check for symmetry, replace \(x\) with \(-x\) in the function and simplify. If \(g(-x) = g(x)\), then the function is even. If \(g(-x) = -g(x)\), then the function is odd. Applying this to our function, we get:\[ g(-x) = -2(-x+1)^2(-x-3)^2 \].The result is not equal to \(g(x)\) or \(-g(x)\) , so this function has no symmetry.

Step 5: Finding the intervals on which the function is positive or negative

Considering the x-intercepts -1 and 3 which divide the x-axis into 3 intervals: (-∞,-1) , (-1,3) , and (3,∞), we pick a test point from within each interval and substitute into the function. If the result is positive, the function is positive within that interval. If the result is negative, the function is negative within that interval.

Step 6: Sketching a graph of the function

Plot the points derived from the analysis including x-intercepts, y-intercept and turns at each x-intercept according to its multiplicity. The end behavior tells that both ends of the graph are pointing downwards.

Key Concepts

End Behavior

When analyzing polynomial functions, understanding the end behavior is fundamental. The end behavior refers to the direction in which the graph of the function proceeds as the value of \( x \) approaches positive or negative infinity. For the function \( g(x) = -2(x+1)^2(x-3)^2 \), the degree is even, and the leading coefficient is negative.

This indicates that both ends of the graph go downward. In simpler terms, as \( x \) moves towards positive infinity, \( g(x) \) tends towards negative infinity. Similarly, as \( x \) approaches negative infinity, \( g(x) \) also heads towards negative infinity.

Understanding the end behavior helps us sketch the graph accurately and predict how the function acts at extreme values of \( x \).

Intercepts

Intercepts are key points where the graph crosses the axes. They give us insights into the roots of the equation and its initial value. There are two primary intercepts to consider: the y-intercept and the x-intercepts.

Firstly, the y-intercept occurs where the graph crosses the y-axis, meaning \( x = 0 \). For \( g(x) \), this is calculated as \( g(0) = -2(0+1)^2(0-3)^2 = 18 \). Therefore, the y-intercept is at the point \( (0, 18) \).

X-intercepts occur where \( y = 0 \). On solving \( g(x) = 0 \), we find that \( x = -1 \) and \( x = 3 \) are our solutions. These are our x-intercepts, and both have a multiplicity of 2, indicating they touch the x-axis but do not fully cross it. Understanding both intercepts helps in plotting key points on the graph.

Symmetry

Symmetry in polynomial functions shows us if the graph is balanced in any way. There are two classic symmetries: even and odd. An even function will be symmetric about the y-axis; an odd function is symmetric about the origin.

To test symmetry in \( g(x) \), we substitute \( -x \) in place of \( x \) and compare it with \( g(x) \) and \( -g(x) \). However, for \( g(x) = -2(x+1)^2(x-3)^2 \), replacing \( x \) with \( -x \) gives a result that matches neither \( g(x) \) nor \( -g(x) \). Thus, this function has no symmetry.

Recognizing symmetry can simplify graphing since symmetric traits mean less detailed analysis is needed for half the graph.

Positivity and Negativity of Functions

When exploring polynomial functions, determining where the function is positive or negative is essential for understanding its graph. This involves establishing the intervals on the x-axis where the function values are above or below zero.

Using the x-intercepts from earlier, we split the x-axis into intervals: \((-\infty, -1)\), \((-1, 3)\), and \((3, \infty)\). By choosing test points within each interval and substituting them into \( g(x) \), we determine the sign of the function in each range.

For example, if we pick a value like \( x = -2 \) in the interval \((-\infty, -1)\), calculate and observe the sign of the output. This aids in predicting where our graph lies in relation to the x-axis in each segment, providing vital information for sketching the graph accurately.