Problem 48 - Algebra and Trigonometry

Question

In Exercises 41–64, 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 turning points to check whether it is drawn correctly. $$f(x)=x^{4}-6 x^{3}+9 x^{2}$$

Step-by-Step Solution

Answer

The graph has an end behavior as \(x \to ± \infty\), \(f(x) \to +\infty\). It has x-intercepts at 0 and 3 where the graph touches x-axis and turns around, and the y-intercept is at (0, 0). The graph has neither y-axis symmetry nor origin symmetry.

Step 1: Determining the end behavior

Using the Leading Coefficient Test, the end behavior of a function \(f(x) = ax^n + \ldots\) where \(n\) is a positive integer, is determined by the sign of the leading coefficient (a) and the degree (n). For \(f(x) = x^4 - 6x^3 + 9x^2\), the leading coefficient is 1 and the degree is 4. Since the degree is even and the leading coefficient is positive, as \(x \to ± \infty\), \(f(x) \to +\infty\).

Step 2: Find the x-intercepts

The x-intercepts of the graph of a function are the values of x where f(x) = 0. Setting \(f(x) = x^4 - 6x^3 + 9x^2 = 0\), we factor out a \(x^2\) to get \(x^2(x^2-6x+9)=0\). This further factors to \(x^2(x-3)^2=0\). So, x=0 or x=3 are the solutions. Thus, the x-intercepts are 0 and 3. The function touches and turns at both x-intercepts.

Step 3: Find the y-intercept

The y-intercept of the function is the value of \(f(x)\) when \(x=0\). For \(f(x) = x^4 - 6x^3 + 9x^2\), the y-intercept is \(f(0) = 0\). Therefore, the y-intercept is at the point (0, 0).

Step 4: Determine the symmetry

To determine the symmetry of the function, check if \(f(-x) = f(x)\) for y-axis symmetry or if \(f(-x) = -f(x)\) for origin symmetry. For this function, neither equation holds true, so the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Additional points and check for correctness

For a polynomial of degree n, there can be at most n-1 turning points. In this case, the degree of the polynomial is 4, so there could be up to 3 turning points. Checking with additional points like \(x=1, 2, 4, 5\) may help in graphing the function accurately.

Key Concepts

X-intercepts

X-intercepts are key points where the graph intersects the x-axis. This means the value of the function, or output, is zero at these points. For polynomial functions, solving for x-intercepts involves setting the function equal to zero and solving for the variable, typically by factoring or using the quadratic formula.

In the example of the given function, \(f(x)=x^4-6x^3+9x^2\), we set it equal to zero: \(x^4-6x^3+9x^2=0\).
By factoring, we first factor out \(x^2\) to get \(x^2(x^2-6x+9)=0\). This further factors into \(x^2(x-3)^2=0\).
Thus, the solutions are \(x=0\) and \(x=3\). These are our x-intercepts, and depending on their multiplicity, the graph will "touch" but not "cross" the x-axis at these points because the multiplicity of each is even.

• If a factor is squared (like \((x-3)^2\)), the graph touches the x-axis and turns around.
• Being even ensures the graph only "grazes" the axis at this point.

Y-intercepts

Y-intercepts are where the graph intersects the y-axis. This occurs when the x-coordinate is zero. To find the y-intercept of a function, we substitute zero for every x in the equation and solve for the function's value.

For our specific function, \(f(x)=x^4 - 6x^3 + 9x^2\), we calculate the y-intercept by finding \(f(0)\): \(f(0) = 0^4 - 6(0)^3 + 9(0)^2 = 0\).
Hence, the y-intercept is at the point (0, 0).

• Being on both the x-axis and y-axis, the point (0, 0) serves as both an x-intercept and y-intercept in this instance.

Graph Symmetry

Graph symmetry helps to identify predictable patterns and behaviors in the graph of a function. For y-axis symmetry, the graph is mirrored over the y-axis if for all x, \(f(-x) = f(x)\).
If there is origin symmetry, the graph is symmetric around the origin meaning for all x, \(f(-x) = -f(x)\).

Let's test these for our function, \(f(x) = x^4 - 6x^3 + 9x^2\):
Substituting -x for x, we have
\(f(-x) = (-x)^4 - 6(-x)^3 + 9(-x)^2 = x^4 + 6x^3 + 9x^2\).
This is neither equal to \(f(x)\)nor \(-f(x)\).
Therefore, the function has neither y-axis symmetry nor origin symmetry.
This means the graph displays a more traditional polynomial curvature without symmetrical mirroring about the axes.

End Behavior

End behavior describes how the function acts as the input value x approaches positive or negative infinity. This largely depends on the polynomial's degree and the sign of the leading coefficient.

For our function, \(f(x) = x^4 - 6x^3 + 9x^2\): the degree is 4, which is even, and the leading coefficient (1) is positive. Based on the Leading Coefficient Test, if the degree is even and the leading coefficient is positive,

• as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity (the graph rises to the right).
• Similarly, as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity (the graph also rises to the left).

This end behavior indicates the graph will open upwards on both ends, akin to a "U" shape.
Understanding this concept is fundamental in sketching polynomial graphs and predicting their shape.