Question

how do identify the vertex and axis of symmetry of the parabola from a curved line on a graph

Answer 1

Short answer: the vertex is the parabola’s highest (or lowest) point on the curve; the axis of symmetry is the straight line through that vertex that splits the parabola into two mirror-image halves.

How to find them from a graph

1. Visual method
- Look for the single peak (maximum) or valley (minimum) on the curved line. That point is the vertex. Read its coordinates from the grid (h, k).
- The axis of symmetry is the straight line that runs through the vertex and divides the parabola into two equal halves. For the usual up/down parabolas it will be a vertical line x = h. For left/right parabolas it will be a horizontal line y = k.

2. Using symmetric points (useful if vertex not obvious)
- Find two points on opposite sides of the parabola that have the same y-coordinate (same height). The axis is the vertical line whose x-value is the average of those two x-values: x = (x1 + x2)/2. The intersection of that line with the curve is the vertex.
- Similarly, for a parabola opening left/right, find two points with the same x and take the average of their y-values to get y = (y1 + y2)/2.

3. Using x-intercepts (if they exist)
- If the parabola opens up/down and you can read the x-intercepts x1 and x2 from the graph, the axis is x = (x1 + x2)/2. The vertex is at x = (x1 + x2)/2 and at y = the curve’s value there.

4. From the quadratic equation y = ax^2 + bx + c
- Vertex x-coordinate h = -b/(2a). Vertex y-coordinate k = f(h).
- Axis of symmetry is x = h.

5. If the parabola is rotated (its axis is not vertical/horizontal)
- The graph will not be symmetric about a vertical or horizontal line. You need to find the vertex by calculus (set the gradient to zero) or by rotating/completing-the-square methods for the general conic. This is more advanced.

Quick example
- If intercepts are x = 1 and x = 5, axis is x = (1+5)/2 = 3. Vertex is at x = 3; read the y-value at x = 3 on the curve to get (3, k).

If you want, tell me whether the parabola in your graph opens up/down or left/right and give a couple of clear points (or a picture), and I’ll find the vertex and axis precisely.