Question

question 35 · 1 point
determine the interval(s) for which the function shown below is decreasing.
write your response in interval notation using the symbol ∪ when necessary.
(graph of a function with x-axis from -8 to 8 and y-axis from -8 to 8, showing a parabola opening downward with vertex at (3, 3.5) approximately, crossing y-axis at (0, -1) and x-axis at around (0.5, 0) and (6, 0))

Explanation:

Step1: Identify the vertex

The graph is a parabola opening downward, so it has a maximum point (vertex) at \( x = 3 \) (from the graph, the peak is at \( x = 3 \)).

Step2: Determine decreasing interval

A function is decreasing when as \( x \) increases, \( y \) decreases. For a downward - opening parabola, the function decreases to the right of the vertex. So the function is decreasing for \( x > 3 \), which in interval notation is \( (3, \infty) \)? Wait, no, wait. Wait, looking at the graph again, the parabola's vertex is at \( x = 3 \), and then it goes down to the right until it goes to negative infinity? Wait, no, the graph: let's check the x - axis. Wait, the parabola has a vertex at \( (3, 3.5) \) (approx), and then as \( x \) increases from 3 to 8 and beyond, the \( y \) - value decreases. Wait, but also, to the left of the vertex, when \( x \) increases from \( -\infty \) to 3, the function is increasing. So the decreasing interval is where \( x \) is greater than 3. Wait, but let's check the graph again. Wait, the parabola: starts from the bottom left, comes up, reaches the vertex at \( x = 3 \), then goes down to the bottom right. So the function is decreasing on the interval where \( x > 3 \), so in interval notation, \( (3, \infty) \)? Wait, no, wait the graph: when \( x = 3 \), it's the maximum. Then for \( x > 3 \), as \( x \) increases, \( y \) decreases. So the interval is \( (3, \infty) \)? Wait, but let's check the x - intercepts. Wait, the graph crosses the x - axis at some point, but the key is the vertex. Wait, maybe I made a mistake. Wait, the vertex is at \( x = 3 \), so the function is decreasing for \( x \in (3, \infty) \)? Wait, no, wait the graph: let's look at the coordinates. The vertex is at \( (3, 3.5) \) (let's say). Then, when \( x = 4 \), \( y \) is less than 3.5, when \( x = 5 \), less than at \( x = 4 \), etc. And as \( x \) goes to the right of 3, \( y \) decreases. So the interval where the function is decreasing is \( (3, \infty) \)? Wait, no, wait the left side: when \( x \) is less than 3, as \( x \) increases, \( y \) increases (since it's coming from the bottom left to the vertex). When \( x \) is greater than 3, as \( x \) increases, \( y \) decreases. So the decreasing interval is \( (3, \infty) \)? Wait, but let's check the graph again. Wait, the parabola: the vertex is at \( x = 3 \), so the function is decreasing on \( (3, \infty) \).

Wait, no, wait maybe I messed up. Wait, the standard form of a parabola \( y = ax^{2}+bx + c \) with \( a<0 \) has vertex at \( x=-\frac{b}{2a} \). In this graph, the vertex is at \( x = 3 \), so the function is decreasing for \( x>-\frac{b}{2a}=3 \), so the interval is \( (3, \infty) \).

Answer:

\( (3, \infty) \)