Problem 50 - 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. $$ f(x)=-2 x^{4}+2 x^{3} $$

Step-by-Step Solution

Answer

The end behavior of \(f(x) = -2x^4 + 2x^3\) is: as x approaches positive or negative infinity, y approaches positive infinity. The \(x\)-intercepts are \(x = 0, 1\) where the graph crosses the x-axis. The \(y\)-intercept is at the origin (0,0). The function has neither \(y\)-axis symmetry nor origin symmetry. For a precise graph, plot few more points on the function if needed.

Step 1: Identify the end behavior

The Leading Coefficient Test can be used to determine the end behavior of the given function \(f(x) = -2x^4 + 2x^3\). Observing the highest power term (the leading term), it's -2x^4. The power is even and the coefficient is negative. For polynomials with an even degree and negative leading coefficient, as x approaches positive or negative infinity, y approaches positive infinity.

Step 2: Find x-intercepts

To find the x-intercepts of the function, we set \(f(x) = 0\) and solve for \(x\). Thus, \(0 = -2x^4 + 2x^3\). We can factor out \(2x^3\), obtaining \(2x^3(-x + 1) = 0\). Consequently, we find the roots \(x = 0, 1\). These are the x-intercepts and the graph crosses the x-axis at these points since the corresponding powers in the original function are odd.

Step 3: Find y-intercept

To find the y-intercept of the function, we set \(x = 0\) and solve for \(f(x)\). For \(f(x) = -2x^4 + 2x^3\), when \(x = 0\), \(f(x) = 0\). So, the y-intercept is (0,0).

Step 4: Determine the symmetry

For \(y\)-axis symmetry, function needs to satisfy f(-x) = f(x), and for origin symmetry, f(-x) = -f(x). If we replace \(x\) with \(-x\) in the original function, we get \(f(-x) = -2(-x)^4 + 2(-x)^3 = -2x^4 - 2x^3\). This is not equal to \(f(x)\) nor \(-f(x)\), thus the function has neither \(y\)-axis symmetry nor origin symmetry.

Step 5: Graph the function

By plotting the intercepts and noting the end behavior and symmetry, one can roughly sketch the function. To confirm, we can plot a few additional points by substituting chosen \(x\) values in the function. The maximum number of turning points a polynomial function can have is one less than the degree of the function. Our function is of degree 4, making the maximum number of turning points 3.

Key Concepts

Leading Coefficient Test

Understanding the Leading Coefficient Test is vital for predicting the end behavior of a polynomial graph. In simple terms, the leading coefficient is the number in front of the term with the highest power in your polynomial. When the degree (the highest power of the variable) is even, and the leading coefficient is negative, like in our example of the function f(x) = -2x^4 + 2x^3, the graph will fall on both ends as x moves towards positive or negative infinity. Conversely, if the leading coefficient were positive, the graph would rise on both ends.

Imagine the leading coefficient as the heaviest part of your graph, pulling it towards infinity (if positive) or negative infinity (if negative). The fact that the degree is even means that the graph will be symmetrical along the vertical axis, and the behavior will be the same in both directions from the y-axis.

x-intercepts of Polynomial

The x-intercepts, also known as zeros or roots, are the points where the graph crosses or touches the x-axis. To find these intercepts, we set the whole polynomial equal to zero and solve for x. With our function f(x), we found the intercepts to be at x = 0 and x = 1. An interesting feature of polynomial graphs is how they behave at these intercepts. Because the factors corresponding to the intercepts have odd powers, the graph will cross the x-axis at these points. This behavior is crucial for sketching accurate graphs of polynomial functions.

y-intercept of Polynomial

The y-intercept is relatively easier to determine. It's the point where the graph meets the y-axis, usually where x is zero. In our polynomial, setting x to zero instantly zeroes the entire function, leaving us with the y-intercept at (0,0). The y-intercept serves as a helpful starting point when plotting a graph manually as it provides a certain point through which the graph will pass.

Polynomial Symmetry

Polynomials may display symmetry, which can simplify graphing considerably. When a polynomial is symmetric about the y-axis, it implies that for every point (x,y), a mirror point (-x,y) also exists on the graph. This occurs if every term in the polynomial has an even power or is a constant. Origin symmetry means that for every point (x,y), the point (-x,-y) is also on the graph, which occurs when the signs of all terms alternate with each successive power.

Our function, f(x) = -2x^4 + 2x^3, however, lacks both y-axis and origin symmetry due to the mixture of even and odd powers and their coefficients. Symmetry can be a powerful tool for graphing but only applies to specific polynomials.

Polynomial End Behavior

The end behavior of a polynomial reveals how the graph behaves as x approaches infinity or negative infinity. In essence, it tells us the final direction of those 'arms' of the graph. For polynomials with an even degree, such as our example f(x), the end behavior is usually the same: both arms will either rise or fall together based on the sign of the leading coefficient.

In our case, with a negative leading coefficient, the polynomial's graph falls off to infinity in both directions, which is a characteristic pattern students can rely on when visualizing polynomial behavior in their graphing exercises.