Questions: Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function. (5 points) f(x)=2 x^2-3 x+5 a. Falls to the left, rises to the right b. Falls to the left, falls to the right c. Rises to the left, rises to the right

Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function. (5 points)
f(x)=2 x^2-3 x+5
a. Falls to the left, rises to the right
b. Falls to the left, falls to the right
c. Rises to the left, rises to the right

Transcript text: 6. Select the correct description of right-hand and left-hand behavior of the graph of the polynomial function. (5 points) \[ f(x)=2 x^{2}-3 x+5 \] a. Falls to the left, rises to the right b. Falls to the left, falls to the right c. Rises to the left, rises to the right

Solution

Solution Steps

To determine the right-hand and left-hand behavior of the graph of a polynomial function, we need to consider the leading term of the polynomial. The leading term of the polynomial \( f(x) = 2x^2 - 3x + 5 \) is \( 2x^2 \). Since the coefficient of \( x^2 \) is positive and the degree of the polynomial is even, the graph will rise to the left and rise to the right.

Step 1: Identify the Polynomial

The given polynomial function is

\[ f(x) = 2x^2 - 3x + 5 \]

Step 2: Determine the Leading Term

The leading term of the polynomial is \(2x^2\). The degree of this polynomial is \(2\), which is even.

Step 3: Analyze the Leading Coefficient

The leading coefficient is \(2\), which is positive. For polynomials with an even degree and a positive leading coefficient, the behavior of the graph is such that it rises on both ends.