Problem 17 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(2 x-1)(x-2)(x-3) $$

Step-by-Step Solution

Answer

Plot zeros at \(x = \frac{1}{2}, 2, 3\) and the y-intercept at \( (0, -6) \). Graph decreases on the left and increases on the right.

Step 1: Identify the Zeros

To find the zeros of the function \(f(x) = (2x - 1)(x - 2)(x - 3)\), set each factor equal to zero: \(2x - 1 = 0\), \(x - 2 = 0\), and \(x - 3 = 0\). Solve these equations to obtain the zeros: \(x = \frac{1}{2},\ x = 2,\ x = 3\).

Step 2: Determine the End Behavior

The polynomial is a cubic function, which means it is of odd degree. The leading term is \(2x^3\) since when expanded the leading term comes from multiplying the highest power terms together: \(2x \cdot x \cdot x\). For odd-degree polynomials with a positive leading coefficient, the graph decreases on the left and increases on the right.

Step 3: Find the Y-Intercept

To find the y-intercept, substitute \(x = 0\) into the function: \(f(0) = (2 \cdot 0 - 1)(0 - 2)(0 - 3) = (-1)(-2)(-3)\). Simplifying gives the y-intercept as \(-6\).

Step 4: Plot the Points

Plot the zeros \(x = \frac{1}{2}, 2,\) and \(3\) on the x-axis. Plot the y-intercept at \( (0, -6) \).These points are key to sketching the function.

Step 5: Sketch the Graph

Use the information from the previous steps to sketch the graph. Start at the y-intercept, move through the zeros from left to right, using smooth curves. The graph approaches positive infinity as \(x\) approaches positive infinity and negative infinity as \(x\) approaches negative infinity, in line with the end behavior determined in Step 2.

Key Concepts

Finding Zeros of a Function

To determine the zeros of a polynomial function, you need to set the equation equal to zero and solve for the values of the variable. In our example, the function is expressed as a product of factors:

• \( f(x) = (2x - 1)(x - 2)(x - 3) \).

To find its zeros, each factor should be individually set to zero. For the factor

• \(2x - 1 = 0\), you solve and find \(x = \frac{1}{2}\).
• Similarly, set \(x - 2 = 0\) to find \(x = 2\) and \(x - 3 = 0\) to find \(x = 3\).

These solutions \(x = \frac{1}{2}, 2,\) and \(3\) are the zeros of the function. These zeros are the x-values where the graph of the function will intersect the x-axis. Finding these intersection points is crucial because they give you key points when sketching the graph of the polynomial function.

End Behavior of Polynomials

Understanding the end behavior of a polynomial is important when graphing the function, as it tells you how the graph behaves as it approaches infinity in either direction. The end behavior is largely determined by the degree of the polynomial and the leading coefficient. When examining our function,

• \( f(x) = (2x - 1)(x - 2)(x - 3) \),
• you expand it to identify the leading term, which is \(2x^3\).

The function is a cubic polynomial, meaning it has an odd degree (3 in this case), and the leading coefficient \(2\) is positive.

• p>A good rule of thumb: for odd-degree polynomials with positive leading coefficients, the graph will fall to the left and rise to the right.

As \(x\) moves towards positive infinity, \(f(x)\) moves upwards towards positive infinity, and as \(x\) moves towards negative infinity, \(f(x)\) moves downwards towards negative infinity. We can use this symmetry to help guide the sketch of the curve.

Y-Intercept of a Function

The y-intercept of a polynomial function is the point where the graph crosses the y-axis. It occurs when all the x-values are zero. Reading off this point from the function itself is straightforward. For the function

• \( f(x) = (2x - 1)(x - 2)(x - 3) \),

substituting \(x = 0\), you'll compute the y-value as follows:

• \( f(0) = (2 imes 0 - 1)(0 - 2)(0 - 3) \),
• which simplifies to \( (-1)(-2)(-3) = -6 \).

Thus, the y-intercept of this function is at the point \((0, -6)\). Plotting this point provides a crucial anchor for drawing the graph and understanding its position and orientation relative to the axes. Alongside the zeros and end behavior, the y-intercept forms part of the backbone of plotting polynomial functions graphically.