Use The Leading Coefficient Test To Determine The End Behavior Of The Graph Of The Given Polynomial Function. Then Use This End Behavior To Match The Polynomial Function With Its Graph. [The Graphs Are Labeled (A) Through (D).] F(X) = -X^3 + X^2+2x F(X) = X^6-6x^4+9x^2 F(X) = X^5-5x^3+4x F(X) = -X^4+1
The Correct Answer and Explanation is :
To analyze the end behavior of a polynomial function using the Leading Coefficient Test, we first focus on the highest-degree term and its coefficient. The leading term dominates the behavior of the function as ( x ) approaches positive and negative infinity. Here’s how to apply this to the given functions:
1. ( f(x) = -x^3 + x^2 + 2x )
- The leading term is ( -x^3 ), where the degree is 3 and the coefficient is negative.
- Degree: Odd (3), so the end behavior will be opposite at the two extremes.
- Leading Coefficient: Negative, meaning as ( x \to \infty ), ( f(x) \to -\infty ), and as ( x \to -\infty ), ( f(x) \to \infty ).
2. ( f(x) = x^6 – 6x^4 + 9x^2 )
- The leading term is ( x^6 ), where the degree is 6 and the coefficient is positive.
- Degree: Even (6), so the end behavior is the same on both sides.
- Leading Coefficient: Positive, meaning as ( x \to \infty ), ( f(x) \to \infty ), and as ( x \to -\infty ), ( f(x) \to \infty ).
3. ( f(x) = x^5 – 5x^3 + 4x )
- The leading term is ( x^5 ), where the degree is 5 and the coefficient is positive.
- Degree: Odd (5), so the end behavior will be opposite at the two extremes.
- Leading Coefficient: Positive, meaning as ( x \to \infty ), ( f(x) \to \infty ), and as ( x \to -\infty ), ( f(x) \to -\infty ).
4. ( f(x) = -x^4 + 1 )
- The leading term is ( -x^4 ), where the degree is 4 and the coefficient is negative.
- Degree: Even (4), so the end behavior is the same on both sides.
- Leading Coefficient: Negative, meaning as ( x \to \infty ), ( f(x) \to -\infty ), and as ( x \to -\infty ), ( f(x) \to -\infty ).
Summary of End Behaviors:
- ( f(x) = -x^3 + x^2 + 2x ): Ends go to ( \infty ) on the left and ( -\infty ) on the right.
- ( f(x) = x^6 – 6x^4 + 9x^2 ): Both ends go to ( \infty ).
- ( f(x) = x^5 – 5x^3 + 4x ): Ends go to ( -\infty ) on the left and ( \infty ) on the right.
- ( f(x) = -x^4 + 1 ): Both ends go to ( -\infty ).
You would match each polynomial to a graph based on these behaviors:
- A polynomial with opposite ends should match a graph where the left goes up and the right goes down.
- A polynomial with both ends going up should match a graph where both ends go upwards.
- A polynomial with both ends going down should match a graph where both ends go down.