Problem 62 - Algebra and Trigonometry

Question

In Exercises 41–64, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-3 x^{3}(x-1)^{2}(x+3)$$

Step-by-Step Solution

Answer

The end behavior of the function is down to the left, down to the right. The x-intercepts are at \(x = 0\), \(x = 1\), and \(x = -3\), with the graph crossing the x-axis at \(x = 0, -3\) and touching the x-axis at \(x = 1\). The y-intercept is at \(y = 0\). The graph has no symmetry. The graph, when drawn with additional points, should respect these properties and have at most 5 turning points.

Step 1: Determining the end behavior

The end behavior of a function is determined by the degree and the sign of the leading coefficient. For the function \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\), notice that the largest exponent is 6 (which is even) and the leading coefficient is -3 (which is negative). Thus, as \(x \rightarrow \pm \infty\), \(f(x) \rightarrow -\infty\). So, the graph falls to the left and falls to the right.

Step 2: Finding the x-intercepts

To find the x-intercepts, set \(f(x)\) equal to zero and solve for \(x\). So, \(-3x^3(x - 1)^2(x + 3) = 0\). The solutions are \(x = 0\), \(x = 1\), and \(x = -3\). At \(x = 0\) and \(x = -3\), the graph crosses the x-axis since the corresponding factors are of odd power, whereas at \(x = 1\), the graph touches the x-axis and turns around as the corresponding factor is of even power.

Step 3: Finding the y-intercept

The y-intercept is the value of the function at \(x = 0\). Substituting \(x = 0\) into the function \(f(x)\), we find the y-intercept to be \(0\).

Step 4: Determine the graph symmetry

A graph has y-axis symmetry if the function is even, and it has origin symmetry if the function is odd. In this case, the function \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\) is neither even nor odd, because it doesn’t fulfill the criteria for either. Therefore, the graph has no symmetry.

Step 5: Additional points and graphing

In order to get an accurate representation of \(f(x)\), it's good to calculate the function's value at a few more points, particularly between the x-intercepts. For example, at \(x=-2\), the function value is 8. Hence, when graphing the function, make sure the graph passes through these points and respects the end behavior and the x-intercepts. Use the fact that this function, being of degree 6, can have at most 5 turning points to check the correct sketch.

Key Concepts

Leading Coefficient Test

Understanding the Leading Coefficient Test is crucial for predicting the end behavior of polynomial functions. This test relies on two properties of the polynomial: the degree and the leading coefficient. The degree is the highest power of the variable in the function, and the leading coefficient is the constant factor of that term.

For any polynomial, if the degree is even and the leading coefficient is positive, the graph will rise to the left and to the right. Conversely, if the leading coefficient is negative, the graph will fall on both ends. For odd degrees, a positive leading coefficient indicates that the graph will fall to the left and rise to the right, while a negative leading coefficient inverts this behavior, causing the graph to rise to the left and fall to the right. When graphing the function given in the exercise, \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\), we note that the degree is 6 (an even number), and the leading coefficient is -3 (a negative number). Consequently, the graph falls to the left and to the right.

X-intercepts of Polynomials

X-intercepts, also known as zeros, roots, or solutions, are the points where the graph of a polynomial touches or crosses the x-axis. To find these intercepts, you must solve the equation \(f(x) = 0\). Each solution corresponds to an x-intercept.

The nature of the intercept—whether the graph crosses or just touches the x-axis—depends on the multiplicity of the root. If the root’s multiplicity, the power of its factor in the polynomial, is odd, the graph will cross the x-axis. If the multiplicity is even, the graph will touch the x-axis and turn around. In our polynomial, \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\), there are three x-intercepts: at \(x = 0\), \(x = 1\), and \(x = -3\). The graph will cross the x-axis at \(x = 0\) and \(x = -3\) and only touch at \(x = 1\) since the factors of \(x\) and \(x+3\) have odd powers, and the factor of \(x-1\) has an even power.

Y-intercept of a Function

The y-intercept is where a graph intersects the y-axis, which naturally occurs when \(x = 0\). To find the y-intercept of a polynomial function, simply evaluate the function with \(x\) set to zero. This intercept can provide a helpful starting point when sketching the graph of the function.

In the example \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\), solving for \(f(0)\) gives us the y-intercept: \(f(0)=0\). Therefore, our graph will pass through the origin, indicating that the y-intercept is the point \(\(0,0\))\). This information complements the knowledge of x-intercepts and helps us plot the function accurately.

Symmetry of Graphs

Symmetry in graphs simplifies the understanding of polynomial functions. A graph can be symmetric with respect to the y-axis if it is an even function, which means for every \(x\), \(f(x)=f(-x)\). A graph displays origin symmetry if it is an odd function, satisfying \(f(-x)=-f(x)\) for all \(x\). However, a graph may also have no symmetry.

To determine if a graph of the given function has symmetry, check the exponents of each term when the function is expanded. For \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\), the function does not exhibit y-axis or origin symmetry; it does not reflect across the y-axis nor does it rotate around the origin and remain unchanged. This absence of symmetry influences the overall shape and balance of the graph.