Problem 53 - 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. \(f(x)=3 x^{2}-x^{3}\)

Step-by-Step Solution

Answer

The end behavior of the function is that as \(x\) approaches \(-\infty\), \(f(x)\) approaches \(-\infty\), and as \(x\) approaches \(\infty\), \(f(x)\) approaches \(-\infty\). The \(x\)-intercepts are \(0\) and \(3\), and the graph crosses the \(x\)-axis at these points. The \(y\)-intercept is \(0\). The function does not have either y-axis nor origin symmetry. The graph of the function includes these features, with the maximum of 2 turning points.

Step 1 - End Behavior

Begin by identifying the degree and leading coefficient of the polynomial, which are 3 and -1 respectively. The degree is odd and the leading term is negative, which implies that as \(x \to -\infty\), \(f(x) \to -\infty\), and as \(x \to \infty\), \(f(x) \to -\infty\). This satisfies the Leading Coefficient Test.

Step 2 - \(x\)-Intercepts

Next, find the \(x\)-intercepts by setting \(f(x)=0\) and solving for \(x\). This gives us \(0=3x^{2}-x^{3}\), which simplifies to \(x=0, 3\). So, the \(x\)-intercepts are \(0\) and \(3\). Since the multiplicity of these roots is 1, the graph crosses the \(x\)-axis at each intercept.

Step 3 - \(y\)-Intercept

Find the \(y\)-intercept by substituting \(x=0\) into the equation, resulting in \(f(0)=0\). Therefore, the \(y\)-intercept is \(0\).

Step 4 - Symmetry

To determine symmetry, substitute \(-x\) for \(x\) into the function and simplify. This gives us \(f(-x)=-3x^{2}+x^{3}\), which is not equal to \(f(x)\) or \(-f(x)\), so there's no y-axis symmetry or origin symmetry.

Step 5 - Graph

The last step is to plot the function using the information gathered. First, plot the intercepts. Then, plot additional points as needed for accuracy. The function has 2 turning points, so make sure your graph has the correct end behavior and turning points.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps in determining the end behavior of a polynomial function. It's essential to identify the degree of the polynomial and the sign of its leading coefficient. For the polynomial function \(f(x) = -x^3 + 3x^2\), the degree is 3, which is odd, and the leading coefficient is -1, which is negative.

Here's a quick rundown on the test:

• If the degree is even and the leading coefficient is positive, both ends of the graph will go up.
• If the degree is even and the leading coefficient is negative, both ends will go down.
• If the degree is odd and the leading coefficient is positive, the left end will go down and the right end will go up.
• If the degree is odd and the leading coefficient is negative, like in our example, both ends will go down as \(x\) moves towards negative and positive infinity.

Therefore, for \(f(x)=3x^2-x^3\), the test shows that the graph's end behavior is down on BOTH sides as \(x\) approaches \(-\infty\) and \(\infty\).

x-intercepts

To find the \(x\)-intercepts of a polynomial function, set \(f(x) = 0\) and solve for \(x\). This will tell us where the graph crosses the \(x\)-axis. For \(f(x) = 3x^2 - x^3\), set the equation to zero: \(0 = 3x^2 - x^3\).

By factoring, we get \(x^2(3-x) = 0\). This means the solutions are \(x = 0\) and \(x = 3\). These points are where the graph meets the \(x\)-axis.

Predicting the graph's behavior at these intercepts involves understanding multiplicity:

• Since the root \(x=0\) has a multiplicity of 2 (as it appears twice in the factored form), the graph touches or "bounces" off the \(x\)-axis here.
• The root \(x=3\) has a multiplicity of 1, indicating that the graph crosses the \(x\)-axis at this point.

y-intercept

The \(y\)-intercept of a function is the point where the graph crosses the \(y\)-axis, and this occurs when \(x = 0\). To find the \(y\)-intercept of \(f(x) = 3x^2 - x^3\), substitute \(0\) in for \(x\).

Here's the calculation:

• \(f(0) = 3(0)^2 - (0)^3 = 0\)

Thus, the \(y\)-intercept of the graph is \((0, 0)\). This is an important point as it confirms that the graph passes through the origin, which is common for many polynomial functions. Knowing this intercept provides a critical anchor point for sketching the graph.

Symmetry in Graphs

Symmetry in a graph determines how the graph behaves around certain axes or points. For polynomial functions, we often check for y-axis symmetry and origin symmetry.

To examine symmetry for \(f(x) = 3x^2 - x^3\), substitute \(-x\) into the function and simplify:

• \(f(-x) = 3(-x)^2 - (-x)^3 = 3x^2 + x^3\)

Compare this result to \(f(x)\) and \(-f(x)\):

• If \(f(-x) = f(x)\), the graph has y-axis symmetry.
• If \(f(-x) = -f(x)\), the graph has origin symmetry.

For our polynomial, \(f(-x)\) neither equals \(f(x)\) nor \(-f(x)\), indicating the graph has neither y-axis nor origin symmetry. Understanding symmetry helps predict the general appearance of the graph and confirms its alignment or misalignment along axes.