Question

Describing End Behavior and Turns For problems 1 and 2, determine the end behavior, the maximum number of turning points, and the maximum number of real roots of each function.

f(x) = -2x^3 + 6x^2 + 5x
g(x) = 4x^4 – 7x^2 + 8
a) For 1: End behavior is down, 3 turning points, 3 real roots. For 2: End behavior is up, 2 turning points, 4 real roots.
b) For 1: End behavior is up, 2 turning points, 4 real roots. For 2: End behavior is down, 3 turning points, 3 real roots.
c) For 1: End behavior is down, 2 turning points, 2 real roots. For 2: End behavior is up, 3 turning points, 4 real roots.
d) For 1: End behavior is up, 3 turning points, 3 real roots. For 2: End behavior is down, 2 turning points, 2 real roots.

Answer 1

The end behavior for f(x) is down since it is a negative cubic function, with a maximum of 2 turning points and 3 real roots. For g(x), the end behavior is up as it is a positive quartic function, with 3 turning points and a maximum of 4 real roots. Option c correctly describes the properties of both functions.

Step-by-step explanation:

To determine the end behavior, the maximum number of turning points, and the maximum number of real roots of each function, we analyze the given functions f(x) = -2x³ + 6x² + 5x and g(x) = 4x⁴ – 7x² + 8.

Function f(x)

For f(x), the leading coefficient is -2 and the highest power of x is 3. Given that it's an odd power and the coefficient is negative, the end behavior of f(x) is down on the right side and up on the left side. A cubic function has a maximum of 2 turning points and can have up to 3 real roots, which matches the maximum number of turning points plus one, as per the Fundamental Theorem of Algebra.

Function g(x)

For g(x), the leading coefficient is positive and the highest power of x is an even number, 4. This indicates that both ends of the graph point upwards, so the end behavior is up. A quartic function (degree 4) can have a maximum of 3 turning points and up to 4 real roots.

Comparing with the options given, the correct answer is c, which correctly describes the end behavior, turning points, and real roots for both functions: for 1: end behavior is down, 2 turning points, 3 real roots; for 2: end behavior is up, 3 turning points, 4 real roots.