Problem 48 - Algebra and Trigonometry

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 turning points to check whether it is drawn correctly. \(f(x)=x^{4}-6 x^{3}+9 x^{2}\)

Step-by-Step Solution

Answer

End behavior is that \(f(x)\) approaches positive infinity as \(x\) approaches either positive or negative infinity. The \(x\)-intercepts are \(x = 0\) and \(x = 3\), with the graph crossing the \(x\)-axis at these points. The \(y\)-intercept is \(y = 0\). The graph has neither y-axis nor origin symmetry.

Step 1: Determining End Behavior

To determine the end behavior, look at the degree and leading coefficient of the polynomial. Here, the degree is 4 (even) and the leading coefficient is 1 (positive). Therefore, as \(x\) approaches positive or negative infinity, \(f(x)\) also approaches positive infinity.

Step 2: Finding the x-intercepts

To find the \(x\)-intercepts, set \(f(x)\) equal to 0 and solve for \(x\). This gives us \(x^{4}-6 x^{3}+9 x^{2} = 0\). Factoring out \(x^{2}\) we get \(x^{2}(x^{2}-6 x+9) = 0\), which gives \(x = 0\) and \(x = 3\) as solutions. The graph crosses the \(x\)-axis at these intercepts.

Step 3: Finding the y-intercept

The \(y\)-intercept is found by setting \(x=0\) in the function. Substituting we find \(f(0) = 0\), so the \(y\)-intercept is at 0.

Step 4: Checking for Symmetry

Substituting \(-x\) into the function, we get \(f(-x)=(-x)^{4}-6(-x)^{3}+9(-x)^{2}=x^{4}+6 x^{3}+9 x^{2}\), which is not equal to \(f(x)\), so there is no y-axis symmetry. For origin symmetry, \(-f(x)\) should be equal to \(f(-x)\), which is also not true in this case. So, there is no symmetry.

Step 5: Plotting additional points and checking the graph

Plotting the \(x\)-intercepts, \(y\)-intercept, and additional points like \(x=1\) and \(x=2\), we get the shape of the graph and can use the Leading Coefficient Test to verify that it correctly shows the end behavior of reaching positive infinity as \(x\) goes to positive or negative infinity.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a simple way to determine the end behavior of a polynomial function. Here, we observe the leading term, which is the term with the highest degree of the polynomial. In the polynomial \(f(x) = x^4 - 6x^3 + 9x^2\), the leading term is \(x^4\).
The leading coefficient is the coefficient of this term, which in this case is 1.

For polynomial functions:

• If the degree is even and the leading coefficient is positive, as \(x\) approaches infinity in both directions, \(f(x)\) goes towards positive infinity.
• If the degree is odd and the leading coefficient is positive, \(f(x)\) will go towards positive infinity as \(x\) increases, and negative infinity as \(x\) decreases.

In this exercise, since the degree is 4 (even), and the leading coefficient is 1 (positive), the graph will rise to positive infinity as \(x\) moves towards both positive and negative infinity.

x-intercepts

Finding the \(x\)-intercepts of a polynomial function involves setting the function equal to zero and solving for \(x\). For this polynomial, we solve the equation \(x^4 - 6x^3 + 9x^2 = 0\).

To do this easily, factor out \(x^2\), giving us \(x^2(x^2 - 6x + 9) = 0\).
This indicates that the solutions are \(x = 0\) and \(x = 3\).

What does this mean for the graph?

• At \(x = 0\), the graph crosses the \(x\)-axis, because the multiplicity (the power of the factor) is even.
• At \(x = 3\), the graph also crosses the \(x\)-axis. Here, there's easier calculus application as the factor's multiplicity is even as well.

Understanding where the graph will intersect the \(x\)-axis and whether it just touches or crosses can help in accurately sketching its behavior.

y-intercept

The \(y\)-intercept of a graph is the point where the graph intersects the \(y\)-axis. To find this point for a polynomial function, we simply set \(x = 0\) and solve for \(f(x)\).

In the polynomial \(f(x) = x^4 - 6x^3 + 9x^2\), substituting \(x = 0\) gives us \(f(0) = 0^4 - 6\times0^3 + 9\times0^2 = 0\).
Therefore, the \(y\)-intercept is at \(0\).

This means the graph passes through the origin point (0, 0) on the coordinate plane, providing a starting point for plotting the overall graph.

graph symmetry

Graph symmetry in polynomial functions can be of two main types: y-axis symmetry and origin symmetry. Let's check for these symmetries by modifying the input of the function and analyzing the output.

**Y-axis Symmetry:**- Substitute \(-x\) into \(f(x)\), giving us \(f(-x) = (-x)^4 - 6(-x)^3 + 9(-x)^2 = x^4 + 6x^3 + 9x^2\). If this equals the original \(f(x)\), the graph has y-axis symmetry. However, since \(f(-x) eq f(x)\), it does not have y-axis symmetry.

**Origin Symmetry:**- For origin symmetry, \(-f(x)\) must equal \(f(-x)\). Testing shows \(-f(x)\) as \(-x^4 + 6x^3 - 9x^2\), which is not equal to \(f(-x)\). Therefore, the graph lacks origin symmetry too.

This careful substitution helps in understanding the behaviors and possible symmetries of the polynomial graph.

polynomial end behavior

Understanding the end behavior of polynomial functions is crucial for accurately sketching graphs. End behavior describes how the polynomial behaves as \(x\) approaches positive or negative infinity.

This end behavior aligns with the Leading Coefficient Test, where we identified:

• The degree of the polynomial is 4 (even).
• The leading coefficient is positive.

Because of these factors, as \(x\) approaches both positive and negative infinity, \(f(x)\) will increase towards positive infinity. This awareness is useful for predicting trends of the polynomial and assisting in the sketching process, ensuring the graph continuously approaches infinity in the upper direction on both ends of the x-axis.