Question

describe the end behavior of the quadratic function graphed below. pick all that apply.
draw
the function values increase as x approaches positive infinity.
the function values decrease as x approaches positive infinity.
the function values increase as x approaches negative infinity.
the function values decrease as x approaches negative infinity.

Explanation:

Step1: Analyze the parabola's direction

The quadratic function's graph is a parabola opening downward (since it has a maximum point at the vertex). The general form of a quadratic is \( y = ax^2 + bx + c \), and if \( a < 0 \), it opens downward.

Step2: Analyze as \( x \to +\infty \)

For a downward - opening parabola, as \( x \) approaches positive infinity (\( x\to+\infty \)), the \( y \) - values (function values) decrease. So "The function values decrease as \( x \) approaches positive infinity" is correct.

Step3: Analyze as \( x \to -\infty \)

For a downward - opening parabola, as \( x \) approaches negative infinity (\( x\to-\infty \)), the \( y \) - values (function values) decrease? Wait, no. Wait, the vertex is at \( x = 0 \). For a downward - opening parabola \( y=ax^{2}+bx + c,a < 0 \), the left - hand side (as \( x\to-\infty \)): let's take the derivative (or just analyze the shape). The parabola opens downward, so as \( x\) moves from \( -\infty \) towards the vertex (\( x = 0 \)), the function values increase, and as \( x\) moves from the vertex towards \( +\infty \), the function values decrease. Wait, I made a mistake earlier. Let's re - analyze. The vertex is at \( (0,0) \) (wait, no, looking at the graph, the vertex is at \( (0,0) \)? Wait, the graph is a parabola opening downward with vertex at \( (0,0) \)? Wait, no, the graph is below the x - axis? Wait, the graph is a parabola opening downward, with vertex at \( (0,0) \)? Wait, no, the graph is drawn with the vertex at the origin (0,0) and opening downward. So the equation is \( y=-ax^{2},a>0 \). So as \( x\to+\infty \), \( y=-ax^{2}\to-\infty \) (decreasing), and as \( x\to-\infty \), \( y = - ax^{2}\to-\infty \)? Wait, no, that can't be. Wait, no, the graph in the picture: let's look at the grid. The vertex is at (0,0), and the parabola opens downward. So when \( x \) is positive and getting larger (approaching \( +\infty \)), the \( y \) - values (function values) decrease (because it's opening downward). When \( x \) is negative and getting more negative (approaching \( -\infty \)), the \( y \) - values (function values) decrease? Wait, no, that's not right. Wait, for \( y=-x^{2} \), when \( x = 1 \), \( y=-1 \); \( x = 2 \), \( y=-4 \) (decreasing as \( x\to+\infty \)). When \( x=-1 \), \( y = - 1 \); \( x=-2 \), \( y=-4 \) (also decreasing as \( x\to-\infty \))? Wait, no, that's because the parabola is symmetric about the y - axis. Wait, but in the graph, the parabola is opening downward, so both as \( x\to+\infty \) and \( x\to-\infty \), the function values decrease? But that contradicts my initial thought. Wait, no, let's check the options. The options are:

1. The function values increase as \( x \) approaches positive infinity.

2. The function values decrease as \( x \) approaches positive infinity.

3. The function values increase as \( x \) approaches negative infinity.

4. The function values decrease as \( x \) approaches negative infinity.

For a downward - opening parabola (\( a<0 \)):

- As \( x\to+\infty \), \( y = ax^{2}+bx + c\to-\infty \) (since \( a<0 \) and \( x^{2}\to+\infty \)), so the function values decrease. So option 2 is correct.

- As \( x\to-\infty \), \( x^{2}\to+\infty \), and since \( a < 0 \), \( y=ax^{2}+bx + c\to-\infty \). But wait, when moving from \( -\infty \) towards 0 (the vertex), the function values increase (because at \( x = -\infty \), \( y\) is a large negative number, and at \( x = 0 \), \( y = 0 \)). Wait, I think I confused the direction. Let's take a point: when \( x=-3 \), \( y=-9 \) (if \( y=-x^{2} \)), when \( x=-2 \), \( y = - 4 \) (increase), when \( x=-1 \), \( y=-1 \) (increase), when \( x = 0 \), \( y = 0 \) (increase), when \( x = 1 \), \( y=-1 \) (decrease), when \( x = 2 \), \( y=-4 \) (decrease), when \( x = 3 \), \( y=-9 \) (decrease). So as \( x\) approaches negative infinity (moving from right to left, going to more negative \( x \)), we are moving from \( x = 0 \) to \( x=- \infty \), so the function values decrease? Wait, no, when \( x\) goes from \( 0 \) to \( - \infty \), \( x\) is becoming more negative, and \( y\) goes from \( 0 \) to \( -\infty \), so the function values decrease as \( x\) approaches negative infinity? But when \( x\) goes from \( - \infty \) to \( 0 \), the function values increase. So the end - behavior (as \( x\to-\infty \)): we look at what happens when \( x\) gets very large in the negative direction. So as \( x\to-\infty \), \( y=-x^{2}\to-\infty \), so the function values decrease as \( x\) approaches negative infinity? Wait, no, that's not correct. Wait, the end - behavior is about what happens as \( x\) approaches \( +\infty \) or \( -\infty \), not as \( x\) approaches the vertex. So for \( y=-x^{2} \):

- As \( x\to+\infty \), \( y\to-\infty \) (function values decrease).

- As \( x\to-\infty \), \( y\to-\infty \) (function values decrease)? But that can't be, because the parabola is symmetric. Wait, no, \( (-x)^{2}=x^{2} \), so \( y(-x)=y(x) \). So as \( x\to+\infty \) and \( x\to-\infty \), \( y\) behaves the same. So both as \( x\to+\infty \) and \( x\to-\infty \), the function values decrease. But let's check the options. The options are:

Option 2: The function values decrease as \( x \) approaches positive infinity. (Correct)

Option 4: The function values decrease as \( x \) approaches negative infinity. (Correct)

Wait, but let's look at the graph again. The graph is a parabola opening downward with vertex at (0,0). So when \( x \) is positive and getting larger (towards \( +\infty \)), the graph goes downward (function values decrease). When \( x \) is negative and getting larger in magnitude (towards \( -\infty \)), the graph also goes downward (function values decrease). So the correct options are the second and fourth.

Answer:

The function values decrease as \( x \) approaches positive infinity, The function values decrease as \( x \) approaches negative infinity (i.e., the options "The function values decrease as \( x \) approaches positive infinity" and "The function values decrease as \( x \) approaches negative infinity" are the correct ones).