Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=x^{3}+3 x^{2}-9 x-27\)

Step-by-Step Solution

Answer

For the polynomial \(f(x) = x^3 + 3x^2 - 9x - 27\), the graph falls to the left and rises to the right (Leading Coefficient Test). The zeros of the polynomial are found by solving \(x^3 + 3x^2 - 9x - 27 = 0\) for \(x\). Plot suitable (x,y) pairs to get solution points. The graph is then sketched by drawing a continuous line through these points.

Step 1: Applying the Leading Coefficient Test

For the function \(f(x) = x^3 + 3x^2 - 9x - 27\), the leading coefficient is 1 (from the term \(x^3\)). The degree of the polynomial is odd, and the leading coefficient is positive. In such case, the graph will fall to the left and rise to the right.

Step 2: Finding Zeros of the Polynomial

To find the zeros of the polynomial, set \(f(x) = 0\), which gives \(x^3 + 3x^2 - 9x - 27 = 0\). Solve this equation for \(x\) by factoring or using a numerical method such as the quadratic formula or synthetic division. The solutions are the x-coordinate points where the graph crosses the x-axis.

Step 3: Plotting Solution Points

To plot solution points, choose a suitable range of x-values and substitute them into the function to find their corresponding y-values. These (x,y) pairs are your solution points.

Step 4: Drawing a Continuous Curve

Use the information from the first three steps to sketch your graph. Plot the zeros on the x-axis, remember the end behavior of the graph as shown by the Leading Coefficient Test, and plot the solution points. Then, draw a smooth, continuous line through these points.

Key Concepts

Leading Coefficient Test

Understanding the Leading Coefficient Test is crucial when analyzing the behavior of polynomial graphs, especially at their extremities. For the polynomial function given: \[ f(x) = x^3 + 3x^2 - 9x - 27 \] the leading term is \(x^3\), which has a coefficient of 1. This coefficient is called the leading coefficient. The degree of the polynomial is 3, which is odd, and the leading coefficient is positive. Knowing these details helps us anticipate the general direction of the ends of the graph.

• If the degree is odd and the leading coefficient is positive, then the graph falls to the left and rises to the right.

This pattern gives us a useful picture of how the graph behaves outside the range of plotted points, assisting us in making a smooth and accurate sketch.

Finding Zeros

Zeros of a polynomial function are the points where the graph intersects the x-axis. For the polynomial: \[ x^3 + 3x^2 - 9x - 27 = 0 \] you can find the zeros by setting the function equal to zero and solving for \(x\). There are a few methods to do this, such as:

• Factoring the polynomial, if possible.
• Using synthetic division.
• Applying the quadratic formula, although this is a cubic polynomial, so more advanced techniques may be needed.

Once you solve the equation, the solutions represent x-values where the graph crosses the x-axis. These zeros are essential for sketching, as they act as anchor points for the curve.

Graph Sketching

Sketching a graph of a polynomial function involves combining several concepts for accuracy. Using the Leading Coefficient Test, zeros, and calculated points help create a holistic view of the graph's shape. After determining the zeros and understanding the end behavior, the next step is to visually interpret these aspects in a sketch. Begin by marking the zeros on the x-axis, noting where the polynomial intersects. Consider the end behavior defined by the Leading Coefficient Test. Here, the polynomial falls to the left and rises to the right, so the ends should reflect this movement. Use the zeros and other key points to guide your pen as you draft a smooth curve, ensuring continuity throughout. Remember, the graph should align with both the zeros and the anticipated end behavior to reflect the true nature of the polynomial function.

Plotting Points

Plotting points between the zeros provides insight into the finer details of the graph's shape. Start by selecting a range of x-values, including those near the zeros, and substitute them into the function \(f(x)\). This will give you corresponding y-values, resulting in pairs like \((x, f(x))\). Plot these points on a coordinate plane to provide more structure to your sketch. This serves to highlight curvature, peaks, and troughs within the graph. As you plot and connect these points, ensure to maintain the continuous nature of the polynomial graph. The combination of these plotted points with the zeros and end behavior allows for a detailed, accurate graph sketch that reflects the entire function accurately.