Question

select the locations of the zeros of function f on the coordinate plane. then select the end behavior of its graph.
as x approaches positive infinity, f(x) approaches negative infinity.
as x approaches negative infinity, f(x) approaches negative infinity.
as x approaches negative infinity, f(x) approaches positive infinity.

Explanation:

Step1: Identify Zeros

Zeros of a function are where \( f(x) = 0 \), i.e., the graph intersects the \( x \)-axis. From the grid, assume the graph (if visible) crosses \( x \)-axis at \( x = -3 \), \( x = 1 \), \( x = 5 \) (example, but need actual graph). Wait, the given graph seems incomplete? Wait, the user's graph: maybe a cubic? Wait, the end behavior options: "As \( x \to +\infty \), \( f(x) \to -\infty \)" and "As \( x \to -\infty \), \( f(x) \to +\infty \)" (if odd degree, leading coefficient negative) or vice versa. Wait, the problem says "select the locations of the zeros" and end behavior. But the graph provided has a grid, maybe the zeros are at \( x = -3 \), \( x = 1 \), \( x = 5 \) (assuming), and end behavior: if leading term is negative odd, then as \( x \to +\infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to +\infty \). But since the user's graph might have a function, let's assume:

Step2: End Behavior

For a polynomial, end behavior depends on degree (odd/even) and leading coefficient. If the graph falls to the right (\( x \to +\infty \), \( f(x) \to -\infty \)) and rises to the left (\( x \to -\infty \), \( f(x) \to +\infty \)), that's odd degree with negative leading coefficient. So the end behavior options: "As \( x \) approaches positive infinity, \( f(x) \) approaches negative infinity" and "As \( x \) approaches negative infinity, \( f(x) \) approaches positive infinity".

Answer:

Zeros: ( -3, 0 ), ( 1, 0 ), ( 5, 0 ) (example, need actual graph). End Behavior: As \( x \to +\infty \), \( f(x) \to -\infty \); As \( x \to -\infty \), \( f(x) \to +\infty \) (matching the option "As x approaches positive infinity, f(x) approaches negative infinity" and "As x approaches negative infinity, f(x) approaches positive infinity").

Wait, the user's options for end behavior are three:

1. As x approaches positive infinity, f(x) approaches negative infinity.

1. As x approaches negative infinity, f(x) approaches negative infinity.

1. As x approaches negative infinity, f(x) approaches positive infinity.

So correct end behavior: If the function is a cubic with negative leading coefficient, then:

• As \( x \to +\infty \), \( f(x) \to -\infty \) (first option).

• As \( x \to -\infty \), \( f(x) \to +\infty \) (third option).

Zeros: The x-intercepts (where y=0) on the grid. Suppose the graph crosses x-axis at (-3,0), (1,0), (5,0) (assuming grid lines: each square is 1 unit).

But since the problem is about graph identification (Mathematics - Algebra/Calculus), using Step-by-Step:

Final Answer (assuming zeros at (-3,0), (1,0), (5,0) and end behavior: "As x approaches positive infinity, f(x) approaches negative infinity" and "As x approaches negative infinity, f(x) approaches positive infinity").