Problem 57 - College Algebra

Question

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{2}(x-1)^{3}(x+2) $$

Step-by-Step Solution

Answer

End behavior: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow +\infty, f(x) \rightarrow +\infty\). x-intercepts: 0 (touches and turns), -2 and 1 (crosses). y-intercept: 0. The graph has no symmetry, neither origin nor y-axis.

Step 1: Determine End Behavior

The behavior modification is determined based upon the leading coefficient of a polynomial function. Our function \(f(x) = x^{2}(x - 1)^{3}(x + 2)\) is of odd degree, 6, with a positive leading coefficient. Thus, the end behavior is: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow +\infty, f(x) \rightarrow +\infty\).

Step 2: Find the x-Intercepts

For finding the x-intercepts, set the function equal to zero and solve for \(x\). \(x^{2}(x - 1)^{3}(x + 2) = 0\), we find \(x = 0, 1, -2\). At \(x = 0\), the graph touches the x-axis and turns around, as the power of x is 2 (an even number). At \(x = 1\) and \(x = -2\), the graph crosses the x-axis, as the powers are 3 and 1 respectively (odd numbers).

Step 3: Find the y-Intercept

To find the y-intercept, set \(x = 0\) in the function. Thus we get, \(f(0) = 0*1*2 = 0\), so the y-intercept is 0.

Step 4: Check for Symmetry

The function will have y-axis symmetry if replacing \(x\) with \(-x\) in the functional expression results in the original function, and origin symmetry if it results in the negative of the original function. Here, neither condition is satisfied, so the function has neither y-axis nor origin symmetry.

Step 5: Sketch the graph

Drawing the function's graph can be somewhat subjective when given no tools. A rudimentary guide includes: marking the x and y intercepts (0, -2, 1), drawing curves for each intercept that cross the x-axis or touch and turn around as per the intercepts' rules, and ensuring end behavior.

Key Concepts

End Behavior

The end behavior of a polynomial graph tells us what happens to the values of the function as the input, or x, becomes very large in either the positive or negative direction.

In this exercise, our polynomial function is expressed as \( f(x) = x^2(x-1)^3(x+2) \). The degree of the polynomial is a key factor here. The degree, or the highest power of x, influences the direction the ends of the graph will point. For our function, the degree is calculated by adding together the exponents: 2 + 3 + 1 = 6, an even number.

Additionally, the leading coefficient (the coefficient of the term with the highest degree) is positive. This tells us that both ends of the plot tend to infinity as x approaches either positive or negative infinity. Hence, the graph falls down to negative infinity as x moves left and rises up to positive infinity as x moves right.

X-Intercepts

The x-intercepts of a polynomial graph are the points where the graph crosses or touches the x-axis. To find these intercepts, you need to set \( f(x) = 0 \) and solve for x.

For our polynomial \( x^2(x-1)^3(x+2) = 0 \), the solution provides us three values for x, namely 0, 1, and -2. The nature of the intercept depends on the power of each factor:

• At \( x = 0 \), the power is 2 (even), indicating that the graph touches the axis and turns back. It doesn't cross the axis at this point.
• At \( x = 1 \) and \( x = -2 \), the powers are 3 and 1 respectively (both odd). This means the graph crosses the x-axis at both these points.

Understanding how the intercepts behave is critical because it informs how to accurately sketch the curve across these points.

Symmetry

Analyzing the symmetry of a polynomial function gives us insight into the overall shape of its graph. There are two main types of symmetry to check:

1. **Y-axis Symmetry**: A graph has y-axis symmetry when replacing x with -x in the function returns the same function, \( f(-x) = f(x) \).

2. **Origin Symmetry**: This occurs when substituting x with -x yields the negative of the function, \( f(-x) = -f(x) \).

For the function \( f(x) = x^2(x-1)^3(x+2) \), neither of these conditions hold true.

• When replacing x with -x, neither y-axis nor origin symmetry is observed since the transformations do not yield equivalently structured expressions.

It's helpful to recognize how these symmetry features (or their absence) affect the simplicity and predictability of a function's graph.

Y-Intercept

The y-intercept is a crucial point on a graph where the curve crosses the y-axis. It’s the output of the function when the input x is zero. To find the y-intercept for any polynomial, substitute x with 0 in the function.

For our function \( f(x) = x^2(x-1)^3(x+2) \), substituting gives:
\[ f(0) = 0^2(0 - 1)^3(0 + 2) = 0 \]

Thus, the y-intercept is at the origin (0,0). Knowing the y-intercept aids in plotting the graph since it provides a fixed start or endpoint of the polynomial on certain axes, especially useful when confirming its placement relative to other x-intercepts.