To solve this problem, we need to match each polynomial function with its correct graph by looking at the end behavior. The end behavior of a polynomial function tells us what happens to the y-values as x gets very large (positive) or very small (negative).
Let’s go through each function one by one:
1.
f(x) = 6x³ + 1
- This is a cubic function (degree 3), and the leading coefficient is positive (6).
- For cubic functions with a positive leading coefficient:
- As x → ∞, f(x) → ∞ (the graph goes up on the right).
- As x → -∞, f(x) → -∞ (the graph goes down on the left).
- Looking at the graphs, graph
d matches this behavior: it goes down on the left and up on the right.
2.
f(x) = -x⁴ + 1
- This is a quartic function (degree 4), and the leading coefficient is negative (-1).
- For quartic functions with a negative leading coefficient:
- As x → ∞, f(x) → -∞ (the graph goes down on the right).
- As x → -∞, f(x) → -∞ (the graph goes down on the left).
- Looking at the graphs, graph
a matches this behavior: it goes down on both ends.
3.
f(x) = -2x³ + 5x²
- This is a cubic function (degree 3), and the leading coefficient is negative (-2).
- For cubic functions with a negative leading coefficient:
- As x → ∞, f(x) → -∞ (the graph goes down on the right).
- As x → -∞, f(x) → ∞ (the graph goes up on the left).
- Looking at the graphs, graph
b matches this behavior: it goes up on the left and down on the right.
4.
f(x) = 4x⁶ - 3x² + 5x - 2
- This is a sextic function (degree 6), and the leading coefficient is positive (4).
- For even-degree polynomials with a positive leading coefficient:
- As x → ∞, f(x) → ∞ (the graph goes up on the right).
- As x → -∞, f(x) → ∞ (the graph goes up on the left).
- Looking at the graphs, graph
c matches this behavior: it goes up on both ends.
Final Answer:
1-d, 2-a, 3-b, 4-c
Parent Tip: Review the logic above to help your child master the concept of polynomial functions worksheet with answers.
