Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ f(x)=(x+3)^{2}(x-2) $$

Step-by-Step Solution

Answer

The graph intercepts \(x\)-axis at \(-3\) and \(2\), \(y\)-intercept at \(-18\); turns at \(-3\); rises right, falls left.

Step 1: Identify the Intercepts

To find the intercepts of the polynomial function \( f(x) = (x+3)^2(x-2) \), start by finding the \(x\)-intercepts. Set the polynomial equal to zero: \((x+3)^2(x-2) = 0\). The solutions are \(x = -3\) and \(x = 2\), so these are the \(x\)-intercepts. The \(y\)-intercept can be found by evaluating the function at \(x = 0\): \(f(0) = (0+3)^2(0-2) = 9(-2) = -18\). The \(y\)-intercept is \((0, -18)\).

Step 2: Determine the Multiplicities

The multiplicity is related to the power of each factor in the polynomial. For the factor \((x+3)^2\), the root \(x = -3\) has multiplicity 2, meaning the graph touches the \(x\)-axis and turns around at this point. For the factor \((x-2)\), the root \(x = 2\) has a multiplicity of 1, meaning the graph crosses the \(x\)-axis at this point.

Step 3: Analyze the End Behavior

The end behavior of a polynomial is determined by the leading term when the polynomial is expanded. The degree of \(f(x) = (x+3)^2(x-2)\) is 3, which is odd, and the leading coefficient, when expanded, is positive. Therefore, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).

Step 4: Sketch the Graph

Use the information gathered: the \(x\)-intercepts at \(x = -3\) and \(x = 2\), \(y\)-intercept at \((0, -18)\), the turning point at \(x = -3\) due to multiplicity 2, and crossing the \(x\)-axis at \(x = 2\). Start the sketch by plotting these intercepts and noting the behavior at each \(x\)-intercept. Incorporate the end behavior into the sketch, ensuring the graph rises to the right and falls to the left.

Key Concepts

x-intercepts

The x-intercepts of a polynomial function are the points where the graph crosses or touches the x-axis. These are found by setting the function equal to zero and solving for x. For the polynomial function given, \( f(x)=(x+3)^{2}(x-2) \), we set the equation to zero: \((x+3)^2(x-2)=0\).

Solving gives us two solutions: \( x = -3 \) and \( x = 2 \). This means the x-intercepts are at \((-3, 0)\) and \((2, 0)\). Both of these points are where the polynomial connects with the x-axis.

• At \(x = -3\), the graph touches and "bounces off" the x-axis, suggesting a higher multiplicity.
• At \(x = 2\), the graph crosses the x-axis, meaning it simply passes through this point.

y-intercepts

The y-intercept is where the graph crosses the y-axis, which is the point when \(x=0\). To find the y-intercept for the function \( f(x)=(x+3)^{2}(x-2) \), substitute \(x = 0\) into the function:

\( f(0) = (0+3)^2(0-2) = 9(-2) = -18 \).

• This results in a y-intercept at \((0, -18)\).
• The y-intercept reveals the value of the function when the input is zero.

Understanding where the function crosses the y-axis provides valuable information for graphing and interpreting the behavior of the polynomial as it starts its journey across the plane.

Multiplicity

Multiplicity in a polynomial context refers to the number of times a particular root is repeated. It affects the shape of the graph at x-intercepts. For the given polynomial \( f(x)=(x+3)^{2}(x-2) \), we observe:

• The root \(x = -3\) has a multiplicity of 2, due to the squared term \((x+3)^2\).
• A multiplicity of 2 means the graph will touch the x-axis at \(x = -3\) and turn back, creating a tangent-like appearance.
• The root \(x = 2\) has a multiplicity of 1, which means the graph will cross the x-axis at this point.

Recognizing multiplicity helps predict how the graph approaches or leaves the x-axis at each intercept.

End behavior

End behavior describes how the graph of a polynomial function behaves as \(x\) approaches negative infinity or positive infinity. For the polynomial \( f(x)=(x+3)^{2}(x-2) \), the degree is 3 (since it's a third-degree polynomial), and the leading coefficient is positive. This information tells us the end behavior:

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

This means the graph will rise on the right end and fall on the left end, typical of odd-degree polynomials with a positive leading coefficient. Understanding end behavior is crucial when sketching the overall shape of the graph, as it indicates how the polynomial reacts at extremes.