Problem 94 - 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. \(g(x)=\frac{1}{10}\left(x^{4}-4 x^{3}-2 x^{2}+12 x+9\right)\)

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{1}{10}(x^{4}-4 x^{3}-2 x^{2}+12 x+9)\) is a curve passing through the points (-1,0), (1,0), (3,0), and (0, \(\frac{9}{10}\)). It rises to positive infinity as x approaches both negative and positive infinity.

Step 1: Apply the Leading Coefficient Test

The Leading Coefficient Test is used to determine the behavior of the graph of a polynomial function as it approaches negative or positive infinity. The leading term of the provided polynomial function is \(x^4\). Since the coefficient is positive and the degree of the polynomial is even, the graph will rise to positive infinity as x approaches both negative and positive infinity.

Step 2: Find the zeros of the polynomial

The zeros of the polynomial are the x-values where the function equals to 0, i.e. solve the equation \( \frac{1}{10}\left(x^{4}-4 x^{3}-2 x^{2}+12 x+9\right)=0\). Factorizing the equation will give zeros at \(x=-1\), \( x=1\), and \(x=3\). So, we have three intersection points: \((-1,0)\), \((1,0)\), and \((3,0)\)

Step 3: Plot sufficient solution points

In order to sketch the graph, some additional points might be helpful. For example, for \(x=0\), we get \(g(0)=\frac{9}{10}\). So we have a fourth point \((0, \frac{9}{10})\).

Step 4: Draw a continuous curve through the points

After plotting all the points from step 2 and step 3 on a graph, we need to draw a smooth continuous curve passing through these points. Remember, the graph rises to positive infinity as x approaches both negative and positive infinity.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a valuable tool for predicting the end behavior of polynomial graphs. It focuses on the leading term of the polynomial, which in this case is \(x^4\). The degree and sign of the leading coefficient play crucial roles.

• The degree of this polynomial, 4, is even.
• The leading coefficient is positive, as it is \(\frac{1}{10}\).

These characteristics tell us that as \(x\) moves toward both negative and positive infinity, the polynomial's graph will rise toward positive infinity. Essentially, regardless of whether \(x\) grows large in the positive or negative direction, the ends of the polynomial's graph will point upwards.

Zeros of Polynomials

Finding the zeros of a polynomial means identifying the x-values at which the polynomial equals zero. These values correspond to points where the graph crosses the x-axis. For the function \[g(x)=\frac{1}{10}(x^{4}-4x^{3}-2x^{2}+12x+9)\], we solve the equation \[\frac{1}{10}(x^4-4x^3-2x^2+12x+9)=0\].

Through factoring, it can be determined that the zeros are \(x = -1\), \(x = 1\), and \(x = 3\). These zeros provide us with important intersection points of the polynomial graph: \((-1,0)\), \((1,0)\), and \((3,0)\). The graph will touch or cross the x-axis at these points.

Plotting Points

While zeros give us points where the graph meets the x-axis, plotting additional points helps to accurately showcase the shape of the graph. For instance, evaluating \(g(x)\) at \(x=0\) gives \(g(0) = \frac{9}{10}\).

This provides another point on the graph: \((0, \frac{9}{10})\). Having just these few points allows us to start imagining the graph's shape, but in practice, adding a few more points might provide greater clarity. Remember:

• Choose different x-values and calculate corresponding y-values.
• Ensure a range of values around and between known zeros.

This process constructs a better outline of the function's behavior.

Continuous Curve Sketching

Once sufficient points have been plotted on the coordinate plane, the next step is sketching the graph. Using these points, draw a smooth, continuous curve that joins them. This is where creativity meets mathematics. It’s important to connect the dots in a fluid manner that respects the polynomial function's behavior.

• Ensure the curve accurately reflects the zeros, approaching the x-axis at the correct points.
• Make sure the ends of the curve ascend to positive infinity, aligning with the Leading Coefficient Test's conclusion.
• The curve between the zeros often takes on distinct features, dipping and rising in line with evaluated points.

The goal is to produce a coherent sketch that accurately depicts the polynomial's nature. With practice, graphing polynomials becomes an intuitive process informed by a combination of calculated points and the expected shape outlined by these core principles.