Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=3 x^{2}+x-2 $$

Step-by-Step Solution

Answer

The function approaches infinity as \( x \) approaches both positive and negative infinity.

Step 1: Identify the Degree and Leading Coefficient

Look at the given function \( f(x) = 3x^2 + x - 2 \). Determine the degree of the polynomial, which is the highest exponent of \( x \). Here, the degree of the polynomial is 2, as the highest power of \( x \) is 2. The leading coefficient is the coefficient of \( x^2 \), which is 3.

Step 2: Determine the Leading Term

For a polynomial, the leading term is the term with the highest power of \( x \). In this case, the leading term of \( f(x) = 3x^2 + x - 2 \) is \( 3x^2 \). The leading term dictates the end behavior of the function.

Step 3: Assess the Leading Coefficient Effect

Since the leading coefficient is positive (3), the parabola opens upwards. For even degree polynomials like this quadratic (degree of 2), a positive leading coefficient means that as \( x \) approaches infinity, \( f(x) \) approaches infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches infinity as well.

Step 4: Conclude the End Behavior

Now, consolidate the findings: As \( x \to +fty \), \( f(x) \to +fty \), and as \( x \to -fty \), \( f(x) \to +fty \). Thus, the end behavior of the function is the same as that of \( 3x^2 \).

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in mathematics. It refers to the highest power of the variable in the polynomial. For instance, in the function \( f(x) = 3x^2 + x - 2 \), the degree is 2. This is because the term with the highest exponent is \( x^2 \). The degree of a polynomial helps determine various characteristics of the polynomial, such as its end behavior and the number of roots or solutions it can have. Typically, the degree tells you:

• The maximum number of solutions or x-intercepts the polynomial can have.
• The polynomial's graph can have up to (degree - 1) turning points.

In our example, with a degree of 2 (making it a quadratic function), the graph can have up to one turning point.

Leading Coefficient

The leading coefficient is the first coefficient in the polynomial when it's written in standard form, ordered by terms from highest to lowest degree. In \( f(x) = 3x^2 + x - 2 \), the leading coefficient is 3, which corresponds to the term with the highest degree, \( 3x^2 \). The leading coefficient plays a crucial role in influencing the shape and direction of the graph. Specifically:

• If the leading coefficient is positive, the graph opens upwards (for even-degree polynomials) or starts low and ends high (for odd-degree polynomials).
• If it's negative, the graph opens downwards (for even-degree polynomials) or starts high and ends low (for odd-degree polynomials).

Thus, a positive leading coefficient like 3 in our quadratic function means it opens upwards.

Leading Term

The leading term of a polynomial is the term containing the highest power of the variable. It sets the groundwork for understanding the polynomial's behavior, especially its end behavior as the values of x become very large or very small. For \( f(x) = 3x^2 + x - 2 \), the leading term is \( 3x^2 \). This particular term is vital because:

• It determines the general direction of the graph as \( x \to \pm\infty \).
• The end behavior of the polynomial largely resembles that of its leading term, especially for large values of \( x \).

In this function, \( 3x^2 \) tells us that as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to \infty \), exhibiting the behavior of a standard upward-opening quadratic.

Quadratic Function

A quadratic function is a second-degree polynomial function of the form \( ax^2 + bx + c \). In this form, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the example \( f(x) = 3x^2 + x - 2 \):

• It's easily identifiable as a quadratic because the highest power of \( x \) is 2.
• The leading coefficient \( a \) is 3, making the parabola associated with this function open upwards.

Quadratic functions are characterized by their parabolic graphs, which can have a maximum or minimum point called the vertex. Some key features include:

• The symmetry of the graph about a vertical line through the vertex.
• Potential to have either 0, 1, or 2 real roots depending on the discriminant \( b^2 - 4ac \).

In conclusion, the quadratic nature of \( f(x) = 3x^2 + x - 2 \) dictates that its graph is a parabola that opens upwards due to the positive leading coefficient.