Question

identify the vertex and the axis of symmetry for the graph:

1. identify the axis of symmetry (x=?).
2. identify the vertex. is it a minimum or maximum?

Explanation:

1) Axis of Symmetry

Step1: Observe the parabola's symmetry

The parabola is symmetric about a vertical line. From the graph, the vertex lies at \( x = 1 \) (by visually inspecting the midpoint or the vertical line that splits the parabola into two mirror - image halves). The axis of symmetry for a parabola \( y = ax^{2}+bx + c \) is given by the formula \( x=-\frac{b}{2a} \), but from the graph, we can directly see that the vertical line of symmetry is \( x = 1 \).

Step1: Find the vertex coordinates

The vertex is the point where the axis of symmetry intersects the parabola. Since the axis of symmetry is \( x = 1 \), and from the graph, the \( y \) - coordinate of the vertex is the lowest point (since the parabola opens upwards). By looking at the graph, the vertex is at \( (1,-1) \).

Step2: Determine if it's a minimum or maximum

For a parabola \( y = ax^{2}+bx + c \), if \( a>0 \), the parabola opens upwards and the vertex is a minimum. Since the given parabola opens upwards (the two arms of the parabola go upwards), the vertex \( (1,-1) \) is a minimum.

Answer:

\( x = 1 \)

2) Vertex and Min/Max