Problem 62 - 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. \(f(x)=-3 x^{3}(x-1)^{2}(x+3)\)

Step-by-Step Solution

Answer

End behavior: The graph falls to the left and rises to the right. x-intercepts: The graph crosses the x-axis at x=0, x=1, and x=-3. y-intercept: The graph intersects the y-axis at the origin, (0, 0). The graph has neither y-axis symmetry nor origin symmetry.

Step 1: Determine end behavior

The Leading Coefficient Test can be used to determine the end behavior of the graph of the function. The highest degree term in the function is the cubic term, -3x^3, and its leading coefficient is -3. Because the degree is odd and the leading coefficient is negative, according to the Leading Coefficient Test, the graph falls to the left and rises to the right.

Step 2: Find the x-intercepts

The x-intercepts of a function can be found by setting f(x) equal to 0 and solving for x. Therefore, solve the equation \(-3x^3(x-1)^2(x+3)=0\). This produces the solutions x=0, x=1, and x=-3. Because each root is of odd multiplicity, the graph crosses the x-axis at each x-intercept.

Step 3: Find the y-intercept

The y-intercept of a function is the value of f at x=0. Therefore, substitute 0 for x in the function: \(f(0)=-3(0)^3(0-1)^2(0+3)=0\). The graph therefore intersects the y-axis at the point (0, 0).

Step 4: Check for symmetry

Substitute \(-x\) for \(x\) in the function and simplify: \(-3(-x)^3(-x-1)^2(-x+3)\). This does not simplify to the original function or to the negation of the function, so the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Draw the graph

To draw the graph of the function, start by marking the x-intercepts and the y-intercept. The graph crosses the x-axis at \(x=0, x=1\), and \(x=-3, y=0\). With the intercept points and knowing that the graph rises to the right and falls to the left, draw the graph accordingly. And since the highest power of this function is 5, we expect it to have at most 4 turning points.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a crucial tool for understanding the behavior of polynomial functions at their ends, or what mathematicians call the "end behavior" of the graph. This test focuses on two main aspects: the degree of the polynomial and the leading coefficient.

• If the degree of the polynomial is even and the leading coefficient is positive, the ends of the graph will both point upwards.
• If the degree is even and the leading coefficient is negative, both ends will point downwards.
• With an odd degree and a positive leading coefficient, the graph will rise to the right and fall to the left.
• Conversely, if an odd degree polynomial has a negative leading coefficient, like in our exercise, the graph will rise to the left and fall to the right.

Knowing just these two pieces of information can help you sketch the rough outline of the polynomial's graph, saving you time and effort during calculations.

X-Intercepts

X-intercepts are points where the graph crosses the x-axis, meaning the function's output, or y-value, is zero at these positions. To find the x-intercepts for the function \(f(x) = -3 x^{3}(x-1)^{2}(x+3)\), we solve the equation \(f(x) = 0\). This results in the roots of the function: \(x = 0, x = 1,\) and \(x = -3\).
These solutions tell us whether the graph crosses or merely touches the x-axis at these points:

• When the multiplicity of a root is odd, the graph crosses the x-axis.
• If the multiplicity is even, it touches and turns back, like a curve bouncing off a surface.

In this function, every intercept has an odd multiplicity, so the graph crosses at each point, providing critical insights for sketching the graph accurately.

Y-Intercept

Finding the y-intercept is simpler than finding the x-intercepts. You just need to set \(x = 0\) in the polynomial function. For our function \(f(x) = -3 x^{3}(x-1)^{2}(x+3)\), substituting \(x\) with zero gives us \(f(0) = -3(0)^3(0-1)^2(0+3) = 0\).
Thus, the graph touches the y-axis at the point \((0, 0)\) which is not only a y-intercept but also an x-intercept in this scenario. This point provides a vital reference when sketching or checking your graph.

Graph Symmetry

Symmetry in graphs provides valuable information about their visual appearance and behavior. For a polynomial function, there are two main types of symmetry to check: y-axis symmetry and origin symmetry.

• Y-axis symmetry means the graph is a mirror image across the y-axis.
• Origin symmetry indicates the graph is symmetrical when rotated 180 degrees about the origin.

To check for these symmetries, substitute \(-x\) for \(x\) in the polynomial and compare the result with the original function. In our exercise, substituting \(-x\) does not yield the original function nor its negative. Therefore, the graph has neither y-axis symmetry nor origin symmetry. Understanding this helps avoid unnecessary drawing errors.

Turning Points of Polynomial Functions

Turning points on a graph are places where the graph changes direction, shifting from rising to falling or vice versa. They are essential for understanding the overall shape of the graph. The maximum number of turning points of a polynomial function is given by its degree minus one.
For the function \(f(x) = -3 x^{3}(x-1)^{2}(x+3)\), the highest degree is 5, meaning it can have up to 4 turning points. While plotting the graph, identifying these points can be crucial in achieving an accurate representation. They are not always easy to find algebraically or by inspection, hence utilizing technology or a strategically placed set of additional calculated points may assist in the visualization.