Question

the vertex is
a pair of symmetrical points are:
the x - intercepts are:
the y - intercept is:
the minimum y - value is:

Explanation:

1. The vertex is:

Step1: Identify vertex coordinates

The vertex of a parabola is the minimum (since it opens upwards) point. From the graph, the vertex is at (4, -8) (assuming the green dashed line is x=4, and the lowest point on the parabola has y=-8). Wait, maybe better to check the grid. Let's see the parabola's minimum point: looking at the graph, the vertex seems to be at (4, -8)? Wait, maybe the grid: each square is 1 unit. Let's re-examine. The parabola's vertex: the x-coordinate is where the axis of symmetry is (green line), which is at x=4 (since between, say, x=0 and x=8? Wait, the x-intercepts: one at x=1, one at x=7? Wait, maybe I miscalculate. Wait, the y-intercept is at (0, 8)? Wait, no, the graph crosses y-axis at (0, 8)? Wait, the initial graph: the blue parabola starts at (0, 8) (since at x=0, y is 8? Wait, the grid: the y-axis has 5, 10, -5, -10. So each grid square is 1 unit. So at x=0, the y-value is 8 (since from (0,0) up 8 units). Then the parabola goes down to a minimum. The axis of symmetry (green line) is at x=4 (since between x=0 and x=8, midpoint? Wait, if x-intercepts are at x=1 and x=7, then axis of symmetry is at x=(1+7)/2=4. Then the vertex is at (4, -8)? Wait, no, the minimum y-value: looking at the graph, the vertex's y-coordinate is -8? Wait, the grid: from y=0 down to y=-8? Wait, maybe the vertex is (4, -8). Wait, maybe I should look again. Alternatively, maybe the vertex is (4, -8). Let's confirm: the parabola opens upwards, so vertex is minimum. So coordinates (4, -8).

Step2: Confirm

So the vertex is (4, -8).

Step1: Find points equidistant from axis of symmetry

The axis of symmetry is x=4. So take a point on the left, say (0, 8) (y-intercept), and its mirror on the right: distance from x=4 is 4 units (4 - 0 = 4), so right point is 4 + 4 = 8? Wait, no, (0,8) is at x=0, distance from x=4 is 4, so mirror is x=4 + 4 = 8, y=8. So (0,8) and (8,8) are symmetrical.

Step2: Verify

Check if they are equidistant from x=4. 4 - 0 = 4, 8 - 4 = 4. Yes, same y-value. So they are symmetrical.

Step1: Find where y=0

The parabola crosses the x-axis at two points. From the graph, looking at the x-axis (y=0), the points are at x=1 and x=7 (since at x=1, y=0; at x=7, y=0). Let's confirm: distance from axis of symmetry (x=4) is 3 units (4 - 1 = 3, 7 - 4 = 3). So x-intercepts are (1, 0) and (7, 0).

Step2: Confirm

So the x-intercepts are at x=1 and x=7, so coordinates (1, 0) and (7, 0).

Answer:

(4, -8)

2. A pair of symmetrical points are: