Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ g(x)=(x+4)(x-1)^{2} $$

Step-by-Step Solution

Answer

The graph crosses at \(x = -4\), is tangent at \(x = 1\), intercepts at \((0, 4)\), and has end behavior of falling left and rising right.

Step 1: Identify the x-intercepts

To find the x-intercepts, set the polynomial equation \(g(x) = (x+4)(x-1)^2 = 0\). This yields the solutions \(x+4=0\) giving \(x = -4\) and \((x-1)^2 = 0\) giving \(x = 1\). Thus, the x-intercepts are at \((-4, 0)\) and \((1, 0)\).

Step 2: Determine the y-intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into \(g(x)\): \(g(0) = (0+4)(0-1)^2 = 4(1) = 4\). Therefore, the y-intercept is \((0, 4)\).

Step 3: Analyze the multiplicity of the factors

For the factor \((x+4)\), the power is 1, so its root at \(x = -4\) has a multiplicity of 1, indicating it crosses the x-axis at this point. For the factor \((x-1)^2\), the power is 2, so its root at \(x = 1\) has a multiplicity of 2, indicating the graph touches the x-axis at this point and is tangent to it.

Step 4: Determine the end behavior

The polynomial \(g(x) = (x+4)(x-1)^2\) is a cubic polynomial with a leading term of \(x^3\). Since the degree is odd, and the leading coefficient is positive, the graph’s end behavior will be as follows: as \(x \to -\infty\), \(g(x) \to -\infty\), and as \(x \to \infty\), \(g(x) \to \infty\).

Step 5: Sketch the graph

Using the intercepts, multiplicities, and end behavior, sketch the graph. Plot the x-intercepts at \((-4, 0)\) and \((1, 0)\), marking the crossing at \(-4\) and tangency at \(1\). Plot the y-intercept at \((0, 4)\). Indicate that the graph falls as \(x\) goes to \(-\infty\) and rises as \(x\) goes to \(\infty\).

Key Concepts

X-intercepts

To determine the x-intercepts of a polynomial function, set the function equal to zero and solve for the variable. For the polynomial given by \( g(x) = (x+4)(x-1)^2 \), the x-intercepts can be found by setting \( g(x) = 0 \). This equation implies that either \( x+4 = 0 \) or \( (x-1)^2 = 0 \). Solving these equations gives us the solutions \( x = -4 \) and \( x = 1 \), respectively. Therefore, the x-intercepts of the polynomial function are both \((-4, 0)\) and \((1, 0)\). The graph of the function will cross or touch the x-axis at these points.

Y-intercepts

Finding the y-intercept of a polynomial function involves evaluating the function at \( x = 0 \). For our polynomial \( g(x) = (x+4)(x-1)^2 \), plug in zero for \( x \):

• \( g(0) = (0+4)(0-1)^2 \)
• Simplifying gives: \( g(0) = 4 \times 1 = 4 \)

Thus, the y-intercept of the function is at the point \((0, 4)\). This means the graph will intersect the y-axis at this point. Finding y-intercepts is straightforward and involves simple substitution of zero for \( x \).

Multiplicity

Multiplicity refers to how many times a particular root is repeated in a polynomial function. The multiplicity determines how the graph intersects the x-axis at the corresponding x-intercept.

• The root from \( x+4 \) has a multiplicity of 1, indicating that the graph crosses the x-axis at \( x = -4 \).
• The factor \( (x-1)^2 \) implies a root at \( x = 1 \) with a multiplicity of 2, meaning the graph only touches the x-axis and is tangent at that point.

Multiplicity affects the shape of the graph and can reveal whether the graph will pass through or bounce off the x-axis at a given intercept.

End Behavior

End behavior describes how the graph of a polynomial behaves as \( x \) approaches positive or negative infinity. The end behavior of a function is determined primarily by the degree and the leading coefficient of the polynomial. In \( g(x) = (x+4)(x-1)^2 \), the highest degree is 3 (since expanding the factors gives an \( x^3 \) term). Since the degree is odd and the leading coefficient (the coefficient of \( x^3 \)) is positive, the end behavior is as follows:

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

These directions mean that the graph falls to the left and rises to the right, characteristic of cubic polynomials with positive leading coefficients.