Question

the function f(x) is represented by the graph below. over what interval is the graph decreasing? a. for all real numbers b. $x \leq -3$ c. $-\infty \leq x \leq 0$ d. $-3 \leq x \leq 3$

Explanation:

Step1: Recall Decreasing Function Definition

A function is decreasing on an interval if, as \( x \) increases, \( f(x) \) decreases. For a parabola (quadratic function) opening upwards, the vertex is the minimum point. The function decreases to the left of the vertex and increases to the right.

Step2: Identify the Vertex's x - coordinate

From the graph, the vertex (minimum point) of the parabola is at \( x = 0 \) (since it's on the y - axis, \( x = 0 \)).

Step3: Determine the Decreasing Interval

For a parabola opening upwards with vertex at \( x = 0 \), the function is decreasing when \( x \) values are less than or equal to 0 (i.e., \( -\infty\leq x\leq0 \)). Let's analyze the options:

• Option A: The function is not decreasing for all real numbers (it increases for \( x\geq0 \)), so A is wrong.
• Option B: The vertex is at \( x = 0 \), not \( x=- 3 \), so B is wrong.
• Option C: This interval \( -\infty\leq x\leq0 \) matches the decreasing interval of the upward - opening parabola with vertex at \( x = 0 \).
• Option D: The function is decreasing on \( -\infty\leq x\leq0 \) and increasing on \( x\geq0 \), so it's not decreasing on \( - 3\leq x\leq3 \) (it increases for \( x\) from 0 to 3), so D is wrong.

Answer:

C. \( -\infty\leq x\leq0 \)