Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=x^{2}-4 x+1$$

Step-by-Step Solution

Answer

The function \(f(x) = x^2 - 4x + 1\) has a maximum of 1 turning point and 2 real zeros.

Step 1: Determine the Number of Turning Points

The number of turning points of a function is always one less than the degree of the function. Therefore, because our function has a degree of 2, it can have at most 2-1 = 1 turning point.

Step 2: Determine the Number of Real Zeros

The number of real zeros of a function is equal to the degree of the function. Therefore, because our function has a degree of 2, it can have at most 2 real zeros.

Key Concepts

Turning Points in Polynomials

Understanding turning points in polynomials is crucial for analyzing the behavior of their graphs. A turning point of a polynomial function is where the graph changes direction. This could be when the graph transitions from increasing to decreasing or vice versa. Now let's delve into how to determine the number of possible turning points.

In our example, the function is given as f(x) = x^2 - 4x + 1. To determine the potential turning points, we look at the function's degree. The degree of a polynomial is the highest power of x that appears in the polynomial. It tells the maximum number of turning points the graph of the polynomial may have. Specifically, the number of turning points is at most one less than the degree of the polynomial.

Since this is a quadratic function (degree 2), it can have at most one turning point. Remember, while the degree gives the maximum number of turning points, it doesn't tell the exact location or the actual number, just the potential maximum.

Degree of a Polynomial Function

The degree of a polynomial function is a fundamental concept that influences the shape and the total number of turning points and real zeros of the graph. As mentioned, the degree is the highest exponent of the variable x. In the exercise, \(f(x) = x^2 - 4x + 1\), the highest power of x is 2, indicating the polynomial is of the second degree.

A polynomial's degree also suggests how the ends of the graph will behave as x becomes very large or very small. For instance, polynomials of even degrees have ends that go off in the same direction, while odd degrees have ends going off in opposite directions. Understanding the degree of a polynomial helps students foresee the number of turning points and predict the end behavior of the polynomial's graph.

Real Zeros of a Polynomial

Real zeros of a polynomial are the values of x that make the polynomial equal to zero—basically, where the graph intersects the x-axis. Identifying the real zeros is a valuable skill in understanding the function's roots and solving polynomial equations.

For the given function, \(f(x) = x^2 - 4x + 1\), the highest degree is 2, so there can be at most 2 real zeros. This aligns with the Fundamental Theorem of Algebra, which states that a polynomial of degree n will have exactly n roots, or zeros, though some may not be real numbers.

In summary, the degree of the polynomial not only informs us about potential turning points but also sets an upper limit on the number of real zeros. Keep in mind that the actual number may be less, especially when some zeros are complex or repeated, but it can never be more than the degree.