Question

the turning point of the graph could only be:
a:(-3,3) b: (4,-2) c: (3,4) d: (-2,3) e: (-3,-2)

Explanation:

Step1: Analyze the parabola's position

The graph is a parabola opening upwards (since both ends go up). The turning point (vertex) should be in the third quadrant or second quadrant? Wait, looking at the graph: the vertex is to the left of the y - axis (x - coordinate negative) and below the x - axis? Wait no, the graph: the parabola has a minimum point (turning point) to the left of the y - axis (x < 0) and the y - coordinate of the turning point: let's see the options. The parabola's turning point: x is negative (since it's left of y - axis), and y - coordinate: looking at the graph, the turning point is below the x - axis? Wait no, the graph: the parabola crosses the y - axis above the origin, and the minimum point (turning point) is to the left of the y - axis. Let's check the x - coordinate: negative. Now y - coordinate: the turning point is below the x - axis? Wait the options: E is (-3, -2), x=-3 (negative), y = - 2 (below x - axis). Let's check each option:

- Option A: (-3,3): y = 3 (above x - axis), but the turning point in the graph seems to be a minimum, so y should be lower.
- Option B: (4, - 2): x = 4 (positive), but the turning point is left of y - axis (x negative), so B is out.
- Option C: (3,4): x positive, out.
- Option D: (-2,3): y positive, out.
- Option E: (-3, -2): x negative, y negative (below x - axis), which matches the position of the turning point (left of y - axis, below x - axis as per the graph's shape: the parabola has a minimum at x negative, y negative? Wait wait, the graph: the parabola goes down to a minimum, then up. The minimum is to the left of the y - axis, and when x is negative, the y - value at the minimum: let's see the options. E has x=-3, y=-2. Let's confirm: the turning point (vertex) of a parabola opening upwards is the minimum point. The graph shows that the vertex is in the third quadrant (x < 0, y < 0) or second? Wait the x - axis is horizontal, y - axis vertical. The parabola crosses the y - axis above the origin (so when x = 0, y>0). The vertex is to the left of the y - axis (x < 0). Now, the y - coordinate of the vertex: since it's a minimum, and the parabola crosses the y - axis at positive y, the vertex's y - coordinate should be less than that (since it's the minimum). But if the vertex is below the x - axis, then y < 0. So among the options, E: (-3, -2) has x < 0, y < 0, which fits. Let's check the x - coordinate: the vertex is at x negative, and among the options with x negative: A (-3,3), D (-2,3), E (-3, -2). A and D have y positive, which would mean the vertex is above the x - axis, but the parabola crosses the x - axis (since it has two x - intercepts: one left of y - axis, one between y - axis and... Wait no, the graph shows two x - intercepts? Wait the parabola crosses the x - axis once left of y - axis and once between y - axis and... Wait no, the graph as drawn: the parabola comes from the top left, goes down to a minimum, then up, crossing the y - axis above the origin. So it has two x - intercepts: one negative (left of y - axis), one positive (between y - axis and...). Wait, if the vertex is above the x - axis, then the parabola would have no x - intercepts, but here it has two. So the vertex must be below the x - axis (so that the parabola crosses the x - axis twice). So the vertex (turning point) has y < 0, x < 0. So E: (-3, -2) is the only one with x < 0 and y < 0.

Step2: Eliminate other options

- A: y = 3 (positive), so vertex above x - axis, but parabola has two x - intercepts, so vertex must be below x - axis (since if vertex is above, no x - intercepts). So A is out.
- B: x = 4 (positive), vertex is left of y - axis, so B out.
- C: x = 3 (positive), out.
- D: y = 3 (positive), out.
- E: x=-3 (negative), y=-2 (negative), matches the position of the turning point (left of y - axis, below x - axis, so that the parabola crosses the x - axis twice).

Answer:

E: (-3, -2)