Problem 46 - 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. $$ f(x)=-x^{4}+4 x^{2} $$

Step-by-Step Solution

Answer

The end behavior of the function is both ends point downwards. The x-intercepts are -2, 0, 2 and the function touches the \(x\)-axis at these points. The y-intercept is at (0,0). The function is symmetric about the \(y\)-axis. The graph of the function is completed by adding additional points (1,3) and (-1,3).

Step 1: Determine the end behavior

Using the Leading Coefficient Test, given the function is \(f(x)=-x^{4}+4 x^{2}\) which is an even degree polynomial and the leading coefficient (-1) is negative, it can be determined that as \(x \rightarrow \pm \infty, f(x) \rightarrow -\infty\). This is because for negative leading coefficient, the left and right ends of the graph point downwards when the exponent is even.

Step 2: Find the x-intercepts

The \(x\)-intercepts are the values of \(x\) when \(f(x)=0\). So, set \(f(x) = 0: -x^{4} + 4x^{2} = 0\). Given that \(x^{2}(4 - x^{2}) = 0\), we can find the roots \(x = -2, 0, 2\). Since the multiplicity of each root is even, the graph touches the \(x\)-axis and turns around at x = -2, x = 0 and x = 2

Step 3: Find the y-intercept

The y-intercept is the value of function when \(x = 0\). Substituting \(x = 0\) in \(f(x)=-x^{4}+4 x^{2}\), we get \(f(0) = 0\). So, the y-intercept is (0,0)

Step 4: Check for symmetry

Substituting \(-x\) for \(x\) in the equation: \(f(-x)= -(-x)^{4}+4 (-x)^{2}= -x^{4}+4 x^{2}\). Since \(f(-x) = f(x)\), the graph is symmetric about the \(y\)-axis. However, there is no origin symmetry, as \(f(-x) \neq -f(x)\).

Step 5: Finding additional points and graph the function

When \(x=1\), \(f(x) = -1 + 4 = 3\) and when \(x=-1\), \(f(x) = -1 + 4 = 3\). So the additional points are: (1,3), (-1,3), which could be plotted on the graph. Note that the number of turning points is one less than the degree of the polynomial - in this case, 3.

Key Concepts

Leading Coefficient Test

In polynomials, the leading coefficient test is a crucial tool for determining the end behavior of a graph. The leading coefficient is the coefficient of the term with the highest degree. In our function, \[ f(x) = -x^4 + 4x^2 \] The leading term is \( -x^4 \) which makes the leading coefficient \( -1 \). Since the degree of the polynomial is even, we look at the sign of this coefficient:

• If it's positive, both ends of the graph point upwards.
• If it's negative, like here, both ends point downwards.

Hence, as \( x \to \infty \) the graph of \( f(x) \) will approach \( -\infty \), and similarly as \( x \to -\infty \). This tells us the graph will look like an upside-down "U" on both sides.

x-intercepts

Finding the x-intercepts involves setting the function equal to zero: \[ -x^4 + 4x^2 = 0 \]Next, we factor the equation, finding common factors or using formulas. Here, notice \( x^2 \) is a common factor: \[ x^2(4 - x^2) = 0 \]This gives us \( x = 0, -2, \) and \( 2, \) as potential x-intercepts. These intercepts show where the graph touches or crosses the x-axis. Important point: check the multiplicity of the roots.

• If the multiplicity is odd, the graph will cross the x-axis.
• If the multiplicity is even, it will touch and rebound.

Each root here has even multiplicity, meaning the graph touches the axis at \(x = -2, 0, \) and \(2 \) and turns around.

y-intercepts

The y-intercept is found by evaluating the function at \( x = 0 \). In our function: \[ f(0) = -0^4 + 4(0)^2 = 0 \]Thus, the y-intercept is the point \((0,0)\). This is where the graph intersects the y-axis. Unlike x-intercepts, a y-intercept occurs only once per function. It is vital for plotting the graph. In this equation, the origin \((0,0)\) serves as both an x-intercept and a y-intercept, indicating the graph passes through the origin.

Graph Symmetry

Understanding symmetry helps in predicting the graph's shape without plotting numerous points. To check for y-axis symmetry, substitute \(-x\) for \(x\): \[ f(-x) = -(-x)^4 + 4(-x)^2 = -x^4 + 4x^2 \] Since \(f(-x) = f(x), \) the function is symmetric about the y-axis. This tells us that the left side of the graph mirrors the right side. Testing for origin symmetry requires checking if \( f(-x) = -f(x) \). In this problem, \(f(-x) eq -f(x) \), so there is no origin symmetry. Recognizing symmetry helps simplify graphing tasks, confirming that once you know one half of the graph, the other half is its mirror image about the y-axis.