Problem 42

Question

In Exercises $35-50$ a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find $x$ -intercepts by setting $f(x)=0$ and solving the resulting polynomial equation. State whether the graph crosses the $x$-axis, or touches the $x$-axis and turns around, at each intercept. c. Find the $y$ -intercept by setting $x$ equal to 0 and computing $f(0)$ 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 fact that the maximum number of turning points of the graph is $n-1$ to check whether it is drawn correctly. $$f(x)=x^{4}-6 x^{3}+9 x^{2}$$

Step-by-Step Solution

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Answer

The graph of the function $f(x)=x^{4}-6x^{3}+9x^{2}$ has both ends pointing upwards, touches the x-axis and turns around at $x = 3$, crosses the x-axis at $x = 0$ and has the y-intercept at $y = 0$. The graph has neither y-axis symmetry nor origin symmetry and it correctly drawn with three turning points.

1Step 1: Determine End Behavior

From the function $f(x)=x^{4}-6x^{3}+9x^{2}$, the highest power of $x$ is 4 and its coefficient is positive. Using the Leading Coefficient Test, both ends of the graph point upwards as $x \to \pm \infty$.

2Step 2: Find $x$-Intercepts

Set $f(x)$ equal to 0 and solve the equation to find the $x$-intercepts. Thus, $x^{4}-6x^{3}+9x^{2} = 0$ which simplifies to $x^{2}(x^{2}-6x+9) = 0$. Factoring further yields $x^{2}(x-3)^{2}=0$. So, $x = 0$ or $x = 3$. The graph touches the x-axis and turns around at $x = 3$, and crosses the x-axis at $x = 0$.

3Step 3: Compute the $y$-Intercept

Find the $y$-intercept by setting $x = 0$ in the function equation, $f(0)=0^{4}-6*0^{3}+9*0^{2}=0$. So, the $y$-intercept is $0$.

4Step 4: Check for Symmetry

Substitute $-x$ for $x$, you get $-x^{4}-6(-x^{3})+9(-x^{2})$, which is not equal to the original function $f(x)$. Hence, the graph has no y-axis symmetry. Since $-f(x) = -x^{4}+6x^{3}-9x^{2}$ is also not equal to the original function, the graph has no origin symmetry.

5Step 5: Graph the Function and Verify

Plot the given points, apply the end behavior, and the symmetry to sketch the graph. Use the fact the maximum number of turning points of the graph is $n-1 = 4-1 = 3$. As seen in the graph, it is drawn correctly with three turning points.

Key Concepts

Leading Coefficient Testx-interceptsy-interceptSymmetryTurning Points

Leading Coefficient Test

The leading coefficient test is a straightforward way to determine the behavior of a polynomial function as the input values become very large or very small (as $x \to \infty$ or $x \to -\infty$). For the polynomial function $f(x) = x^4 - 6x^3 + 9x^2$, the degree is 4, which is an even number, and the leading coefficient (the coefficient of $x^4$) is positive. This tells us:

The ends of the graph behave similarly. Since the coefficient is positive, both tails of the graph will point upwards.
If the leading coefficient were negative, the ends would both point downwards.

Understanding the end behavior is really helpful when sketching a rough graph of the polynomial.

x-intercepts

Finding the $x$-intercepts of a polynomial means setting the function equal to zero and solving for $x$. For our function $f(x) = x^4 - 6x^3 + 9x^2$, we solve:

$x^4 - 6x^3 + 9x^2 = 0$
Factor to $x^2(x - 3)^2 = 0$

This gives us two solutions: $x = 0$ and $x = 3$.
These intercepts tell us:

The graph crosses the $x$-axis at $x = 0$, which means it passes through this point.
At $x = 3$, the graph touches the axis and turns around, showing a behavior like a bounce; this is due to the squared factor $(x - 3)^2$.

Intercepts help us understand how the graph interacts with the $x$-axis.

y-intercept

The $y$-intercept of a function is found by substituting $x = 0$ in the polynomial. It's where the graph crosses the $y$-axis.
For $f(x) = x^4 - 6x^3 + 9x^2$:

Set $x = 0$, calculate $f(0) = 0^4 - 6 \times 0^3 + 9 \times 0^2$.
This simplifies to $f(0) = 0$.

The $y$-intercept of this polynomial is at the origin (0, 0). This means the graph will touch and cross the $y$-axis at this point.

Symmetry

Checking for symmetry in polynomial functions helps in graphing them quicker and understanding their nature. We generally check for two types of symmetry:

Y-axis symmetry is tested by replacing $x$ with $-x$. If the function $f(-x)$ results in the original $f(x)$, the graph is symmetric about the y-axis. For $f(x) = x^4 - 6x^3 + 9x^2$, substituting $-x$ results in $-x^4 - 6(-x)^3 + 9(-x)^2$, which does not equal $f(x)$. So, no y-axis symmetry.
Origin symmetry is tested by checking if $-f(x) = f(-x)$. Again, this condition does not hold for our polynomial.

Thus, the graph of $f(x)$ has no specific symmetry, making it more unique and interesting to analyze.

Turning Points

A turning point is where the graph changes direction from increasing to decreasing or vice versa. For a polynomial of degree $n$, the maximum number of turning points is $n-1$.
For $f(x) = x^4 - 6x^3 + 9x^2$, the degree is 4, so: