Problem 18 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Graph intersects at (0,0) and (1,0); touches but doesn't cross at (2,0); both ends rise to infinity.

Step 1: Identify the Roots

To graph the polynomial function, first identify the roots by solving the equation \( f(x) = 0 \). Set each factor equal to zero: \( x = 0 \), \( x - 2 = 0 \), and \( x - 1 = 0 \). The roots are \( x = 0 \), \( x = 2 \) (with multiplicity 2), and \( x = 1 \).

Step 2: Analyze End Behavior

Determine the end behavior of the polynomial by considering the degree and leading coefficient. The polynomial is of degree 4 (since \( x^1 \times (x-2)^2 \times (x-1) \) results in \( x^4 \)) and the leading coefficient is positive. Therefore, as \( x \to -\infty \), \( f(x) \to \infty \) and as \( x \to \infty \), \( f(x) \to \infty \).

Step 3: Evaluate Behavior at Each Root

Examine the behavior of the graph at each root:- At \( x = 0 \), the graph crosses the x-axis since the root has an odd multiplicity (1).- At \( x = 1 \), the graph also crosses the x-axis because the root has an odd multiplicity (1).- At \( x = 2 \), the graph touches the x-axis and turns back since the root has an even multiplicity (2).

Step 4: Determine Additional Points

Calculate a few more points to ensure the shape of the graph is correct. Choose points like \( x = -1 \), \( x = 1.5 \), and \( x = 3 \) and substitute them into the function to get the corresponding \( y \)-values, which helps plot the graph more accurately.

Step 5: Sketch the Graph of the Function

Using the roots, end behavior, and additional points, sketch the graph. Start from the left, noting it begins from positive infinity, intersects at \( (0,0) \), crosses again at \( (1,0) \), touches the x-axis at \( (2,0) \) without crossing, and then continues upwards. This forms a curve that reflects the analyzed properties.

Key Concepts

End Behavior

Understanding the end behavior of a polynomial function is crucial for sketching its graph. To determine end behavior, we observe both the degree and the leading coefficient of the polynomial.
For the function \( f(x)=x(x-2)^{2}(x-1) \), we note that when expanded, the leading term is \( x^4 \), making the degree 4.
Since the degree is even and the leading coefficient (1) is positive:

• As \( x \to -\infty \), \( f(x) \to \infty \).
• As \( x \to \infty \), \( f(x) \to \infty \).

This indicates that both ends of the graph shoot upwards towards infinity as you move away from the origin.

Roots of Polynomials

Roots of a polynomial are the values of \( x \) where the function equals zero. These points are where the graph intersects the x-axis. To find the roots, solve for \( f(x) = 0 \).
For \( f(x)=x(x-2)^{2}(x-1) \), setting each factor equal to zero gives the roots:

• \( x = 0 \)
• \( x = 1 \)
• \( x = 2 \)

Roots are essential for starting the sketch of a polynomial's graph because they provide the intersection points on the x-axis.

Graphing Polynomials

Graphing a polynomial involves plotting the x-intercepts and understanding the general shape determined by roots and end behavior. Start by marking the roots on the x-axis:

• Plot \( (0,0) \), \( (1,0) \), and \( (2,0) \)

Consider the calculated end behavior; remember the graph will rise to infinity on both sides.
Determine additional points by calculating values such as \( f(-1), f(1.5), f(3) \) to refine your sketch.
Connect these points smoothly, respecting root behavior (crossing or touching the x-axis) and calculated ends.

Multiplicity of Roots

Multiplicity of a root influences how the graph behaves at that root. It refers to the number of times a specific root is repeated. Analyze our example \( f(x) = x(x-2)^2(x-1) \):

• \( x = 0 \) has multiplicity 1 (graph crosses).
• \( x = 1 \) has multiplicity 1 (graph crosses).
• \( x = 2 \) has multiplicity 2 (graph touches and turns back).

Odd multiplicity means the graph crosses the x-axis. Even multiplicity means it touches and rebounds without crossing. This behavior greatly affects the overall shape of the polynomial's graph.