Question

directions: state the maximum number of turns the graph of each function could make.

1. $f(x) = x^3 - 2x^2 + 4$
2. $f(x) = -x^4 - x^3 + x^2$

Explanation:

Problem 2:

Step1: Recall the rule for turning points

For a polynomial function of degree \( n \), the maximum number of turning points is \( n - 1 \).

Step2: Determine the degree of \( f(x)=x^{3}-2x^{2}+4 \)

The degree of the polynomial is the highest power of \( x \), here the highest power is \( 3 \) (from \( x^{3} \)).

Step3: Calculate the maximum number of turning points

Using the formula \( n - 1 \), where \( n = 3 \), we get \( 3- 1=2 \).

Step1: Recall the rule for turning points

For a polynomial function of degree \( n \), the maximum number of turning points is \( n - 1 \).

Step2: Determine the degree of \( f(x)=-x^{4}-x^{3}+x^{2} \)

The degree of the polynomial is the highest power of \( x \), here the highest power is \( 4 \) (from \( -x^{4} \)).

Step3: Calculate the maximum number of turning points

Using the formula \( n - 1 \), where \( n = 4 \), we get \( 4 - 1=3 \).

Answer:

The maximum number of turns for \( f(x)=x^{3}-2x^{2}+4 \) is \( 2 \).

Problem 3: