Problem 48 - 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. $$ f(x)=x^{4}-6 x^{3}+9 x^{2} $$

Step-by-Step Solution

Answer

The graph has neither y-axis nor origin symmetry, touches the x-axis at \(x=0\) and \(x=3\), and has a y-intercept at 0. Both ends of the graph tend upwards.

Step 1: Determining the End Behavior

We can determine the end behavior of a polynomial function by examining its degree and leading coefficient. In this case, the degree of the polynomial is 4, and the leading coefficient is 1. Since the degree is even and the leading coefficient is positive, both ends of the graph will tend upwards.

Step 2: Finding the x-intercepts

The x-intercepts of a function are found by setting the function equal to zero and solving for \(x\). Thus, \(x^{4}-6 x^{3}+9 x^{2}=0\). Factoring out \(x^2\), we get \(x^{2}(x^{2}-6 x+9) = 0\). This further factors to \(x^{2}(x-3)^{2}=0\). The solutions are \(x=0, 3, 3\), which means the graph touches the x-axis at \(x=0\) and \(x=3\), and turns around at these points.

Step 3: Finding the y-intercept

The y-intercept of a function is found by setting \(x = 0\). Setting \(x = 0\) in the function \(f(x)=x^{4}-6 x^{3}+9 x^{2}\) returns \(f(0) = 0\). Therefore, the y-intercept is 0.

Step 4: Checking for Symmetry

A function has y-axis symmetry if \(f(-x) = f(x)\) and origin symmetry if \(f(-x) = -f(x)\). Testing this function for symmetry gives: \(f(-x) = (-x)^4 - 6(-x)^3 + 9(-x)^2 = x^{4}+6 x^{3}+9 x^{2}\), which is not equal to \(f(x)\), so there is no y-axis symmetry and \(f(-x) = -f(x)\) which is not true as well, so there is no origin symmetry. The graph has neither y-axis nor origin symmetry.

Step 5: Graphing the function

Draw an x-y plane, mark the x and y intercepts found in Steps 2 and 3, respectively. Sketch the curve noting the end behavior from step 1 and the point at which the graph touches the x-axis from step 2. There are no additional points to check and the graph can be drawn correctly with these.

Key Concepts

End Behavior

In polynomial functions, the end behavior is determined by the degree and the leading coefficient. The degree is the highest power of the polynomial, and in our example, it is 4. The leading coefficient is the number in front of the term with the highest degree, which is 1 in this case.

When the degree is an even number and the leading coefficient is positive, both ends of the graph tend to go upwards. This means as the value of the function goes towards positive or negative infinity, the graph rises towards positive infinity. Thus, the graph in our exercise rises on both the left and the right.

Intercepts

Intercepts are points where the graph intersects the axes. They help us understand where the function interacts with the x-axis and y-axis.

The x-intercept occurs where the function equals zero. For our polynomial, we set the equation to zero: \[ x^4 - 6x^3 + 9x^2 = 0 \]By factoring, we determine the x-intercepts to be \(x = 0\) and \(x = 3\). Both these points are where the graph "touches" the x-axis and then turns around. This behavior typically occurs when the factor has even multiplicity, which indicates that at \(x = 3\), the factor \((x-3)^{2}\) is present.

The y-intercept is found by setting \(x = 0\) in the function. Inserting this gives us the value \(f(0) = 0\), confirming the y-intercept is also \(0\).

Symmetry

Symmetry in functions helps identify if a graph reflects uniformly along a certain axis or about a point. There are two primary types of symmetry to consider: y-axis symmetry and origin symmetry.

Y-axis symmetry occurs when the function fulfills \(f(-x) = f(x)\). Upon substitution, our function becomes \[ (-x)^4 - 6(-x)^3 + 9(-x)^2 = x^4 + 6x^3 + 9x^2 \], which is not equal to the original function \(f(x)\). Therefore, it does not have y-axis symmetry.

Origin symmetry is present if \(f(-x) = -f(x)\). When computed, \[ f(-x) eq -f(x) \] for our function. This means there is neither y-axis nor origin symmetry in the polynomial.

Graphing Functions

Graphing functions involves plotting key points and understanding trends in the graph's behavior. You should start by marking the axes and plotting the intercepts found earlier: \((0,0)\) for both the x and y-intercepts, and \((3,0)\) for the x-intercept.

With the intercepts plotted, utilize the determined end behavior to sketch the function: both ends of the graph should rise upwards. Since we've discovered no symmetry, this sketch remains unchanged by reflecting or rotating.

Ensure the graph touches and turns at \(x = 0\) and \(x = 3\) based on the multiplicity of the roots. By taking these steps, you can accurately represent the polynomial function.