Problem 61 - Algebra and Trigonometry

Question

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. \(f(x)=-2 x^{3}(x-1)^{2}(x+5)\)

Step-by-Step Solution

Answer

End behavior: \( \lim_{x \to -\infty} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = \infty \). X-intercepts: 0, 1, -5 where the function crosses the x-axis. Y-intercept: 0. Symmetry: None.

Step 1: Apply the Leading Coefficient Test

The leading coefficient of this polynomial function is -2 and the degree is 5 (odd). When the leading coefficient is negative and the degree is odd, the end behaviour of the graph is: \( \lim_{x \to -\infty} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = \infty \)

Step 2: Find the x-intercepts

Set the function equal to zero and solve for \(x\). Here, \(x=0, x=1\) and \(x=-5\) are the \(x\)-intercepts. Given that all coefficients are of odd powers, the graph crosses the x-axis at each of these intercepts.

Step 3: Find the y-intercept

Set \(x = 0\) in the function to get the y-intercept. The y-intercept is \(f(0)=-2(0^{3})(0-1)^{2}(0+5)=0\).

Step 4: Test for Symmetry

Plug \(-x\) into the function and simplify. If the result is not identical to the original function, then the function is not symmetric to the y-axis. If the result is the negation of the original function, it has origin symmetry. If neither condition is met, the function has no symmetry. For our function, it doesn't have y-axis or origin symmetry.

Step 5: Additional Points and Graph.

If necessary, select a few more points along the x-axis to better visualize the function. For our cubic function, the maximum number of turning points is \(n-1=5-1=4\), where \(n\) is the degree of the polynomial. This can assist in verifying the correct sketch of the function.

Key Concepts

Leading Coefficient Test

Understanding the end behavior of a polynomial graph is essential, and that’s where the Leading Coefficient Test comes in handy. It gives us clues about the function's arrows as we head towards infinity. For example, when dealing with a polynomial where the leading coefficient (the coefficient of the term with the highest power) is negative and the degree is odd, we anticipate certain movement. This scenario implies that the graph falls to the left (\f\((x \to -\text{infinity})\f\)) and rises to the right (\f\((x \to \text{infinity})\f\)), like a swooping flight path of a bird. This concept directly influences our predictions about how the graph behaves beyond the plotted points and provides the framework for the overall shape of the graph.

x-intercepts of a polynomial

The x-intercepts, also known as the zeros of a polynomial, are the points where the graph crosses or touches the x-axis. To find them, we set the polynomial equal to zero and solve for x. The x-intercepts provide a 'footprint' of where our polynomial function will step along the x-axis. Notably, the behavior of the graph at each intercept is also telling: if the power of the factor is odd, the graph will cross the x-axis, while if it's even, the graph merely touches the axis and turns around. This gives us part of the 'skeleton' upon which we'll later flesh out the rest of our graph.

y-intercept calculation

Just as with x-intercepts, the y-intercept is a critical point where the graph of the polynomial meets the y-axis. We find this by simply evaluating the polynomial function at x=0. It's the home base from which we start our graph and provides a reference point for understanding how the function behaves in relation to the y-axis. It’s crucial to understand that there can only be one y-intercept for any polynomial function, as it can only cross the y-axis once.

polynomial symmetry

Symmetry in polynomials can simplify our graphing task significantly. If a polynomial function is symmetric with respect to the y-axis, then for every point (\f\((x, y)\f\)) on the graph, the point (\f\((-x, y)\f\)) will also be on the graph. Similarly, origin symmetry implies that for every (\f\((x, y)\f\)), a corresponding (\f\((-x, -y)\f\)) exists. To check for symmetry, we substitute \f\((-x)\f\) for x. A lack of symmetry doesn't necessarily complicate things; it just means we have to plot more points to determine the shape of the graph.

graphing polynomial functions

Once we have the x-intercepts, y-intercept, and understand the end behavior and symmetry of our polynomial, we can begin to sketch our graph. Start with marking the intercepts and noting the end behavior. Next, we determine how many turns the graph can take—which is always one less than the degree of the polynomial function. These turning points represent local maxima or minima on the graph. More points may be plotted for accuracy, and once we have an outline roughed in with these criteria met, we can connect the dots to visualize the complete journey of our polynomial function.