Problem 264 - Precalculus

Question

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

Step-by-Step Solution

Answer

The graph intersects at \((-3,0)\) and \((2,0)\), rebounding at \(-3\); y-intercept is \((0,-18)\); end behavior is down on the left and up on the right.

Step 1: Identify the x-intercepts

The x-intercepts of a polynomial function occur where the function equals zero. Set the equation to zero: \[ (x+3)^2(x-2) = 0 \]This gives us the solutions:1. \(x = -3\) with a multiplicity of 2,2. \(x = 2\) with a multiplicity of 1.

Step 2: Identify the y-intercept

The y-intercept of the polynomial is the value of the function when \(x = 0\). Substitute \(x = 0\) into the function:\[ f(0) = (0+3)^2(0-2) = 9(-2) = -18 \]So the y-intercept is at \((0, -18)\).

Step 3: Determine the Multiplicity of Roots

The root \(x = -3\) has a multiplicity of 2, which means the graph touches and rebounds off the x-axis at this point. The root \(x = 2\) has a multiplicity of 1, which means the graph crosses the x-axis at this point.

Step 4: Analyze the End Behavior

The polynomial is of degree 3, as the highest degree term is produced by combining the terms \(x^2\) (from \((x+3)^2\)) and \(x\) (from \((x-2)\)), yielding \(x^3\). Since the leading term \(x^3\) is positive and of odd degree, the end behavior will be: - As \(x \to -\infty\), \(f(x) \to -\infty\).- As \(x \to \infty\), \(f(x) \to \infty\).

Key Concepts

Understanding x-intercepts

In polynomial functions, x-intercepts are the points where the graph crosses or touches the x-axis. These occur when the function is equal to zero. For the function \(f(x) = (x+3)^2(x-2)\), we find the x-intercepts by setting the equation to zero: \((x+3)^2(x-2) = 0\). This tells us the solutions are \(x = -3\) and \(x = 2\).
The x-intercept \(x = -3\) occurs twice, so it has a multiplicity of 2. This means the graph touches the x-axis but doesn't cross it here, as we'll see in the multiplicity section later. The x-intercept \(x = 2\) has a multiplicity of 1, meaning it simply crosses the x-axis at this point. Understanding these x-intercepts is essential for sketching the graph of the polynomial.

Discovering y-intercepts

The y-intercept of a polynomial function is where the graph intersects the y-axis. This happens when all other variables are zero, specifically when \(x = 0\). To find this for the polynomial \(f(x) = (x+3)^2(x-2)\), we substitute \(x = 0\) into the function:

• First, compute \((0+3)^2 = 9\).
• Next, calculate \(0-2 = -2\).
• Multiply these results: \(9 \times (-2) = -18\).

Thus, the y-intercept is at the point \((0, -18)\). This tells us the graph of the polynomial passes through this point on the y-axis, which can be helpful when graphing the function.

Exploring multiplicity

Multiplicity in polynomial functions describes how many times a particular root is repeated. It has a direct impact on the graph's behavior at the x-intercepts. In the exercise, we have two roots: \(x = -3\) with a multiplicity of 2 and \(x = 2\) with a multiplicity of 1.

• If a root has an odd multiplicity, the graph will cross the x-axis at that point.
• If a root has an even multiplicity, the graph will touch the x-axis and turn around (bounce off) without crossing it.

For our specific function, at \(x = -3\), the graph does not cross the x-axis but instead touches it and turns around, because the multiplicity is even (2). In contrast, at \(x = 2\), the graph crosses the x-axis, as the multiplicity is odd (1). Understanding multiplicity helps us predict the behavior of the graph at different intercepts.

Analyzing end behavior

The end behavior of a polynomial gives us insight into what happens to the graph as \(x\) reaches very large (positive) or very small (negative) values. This is determined by the polynomial's leading term, which is the term with the highest degree (power of \(x\)).
In our polynomial \(f(x) = (x+3)^2(x-2)\), the highest degree term is produced by multiplying \(x^2\) from \((x+3)^2\) and \(x\) from \((x-2)\), giving us a leading term of \(x^3\). This term is positive (since the coefficient is not negative) and of odd degree (3), guiding our analysis of the end behavior:

• As \(x \to -\infty\), \(f(x) \to -\infty\). The graph will decrease without bound in the negative y-direction.
• As \(x \to +\infty\), \(f(x) \to +\infty\). The graph will increase without bound in the positive y-direction.

Recognizing the end behavior helps in sketching the overall path of the graph and in understanding how the function behaves at the extremes.