Question

using the graph, determine the equation of the axis of symmetry.
answer
answer type: two points
two points: ( ) and ( )

Explanation:

Step1: Identify symmetric points

The parabola intersects the x - axis at \(x=-6\) (point \((-6,0)\)) and \(x = - 2\) (point \((-2,0)\)). These two points are symmetric about the axis of symmetry.

Step2: Calculate the axis of symmetry

The formula for the axis of symmetry (for a parabola with x - intercepts \(x_1\) and \(x_2\)) is \(x=\frac{x_1 + x_2}{2}\). Substituting \(x_1=-6\) and \(x_2=-2\), we get \(x=\frac{-6+( - 2)}{2}=\frac{-8}{2}=-4\). To find two points on the axis of symmetry, we can take any y - value. Let's take \(y = 1\) (from the vertex). So two points on the axis of symmetry \(x=-4\) are \((-4,0)\) and \((-4,1)\) (or other points with \(x=-4\)). But looking at the x - intercepts, the symmetric points on the parabola (other than the vertex) are \((-6,0)\) and \((-2,0)\), and the axis of symmetry passes through the mid - point. The mid - point of \((-6,0)\) and \((-2,0)\) is \((\frac{-6 + (-2)}{2},0)=(-4,0)\). Another point on the axis of symmetry can be the vertex, which is \((-4,1)\) (assuming the vertex has y - coordinate 1 from the graph). But if we consider the two x - intercepts, the axis of symmetry is the vertical line through their mid - point. So two points on the axis of symmetry are \((-4,0)\) and \((-4,1)\) (or any two points with \(x=-4\)). But if we take the two x - intercepts, they are symmetric about \(x = - 4\), so the axis of symmetry is \(x=-4\), and two points on it can be \((-4,0)\) and \((-4,2)\) (or other y - values). But from the graph, the vertex is at \((-4,1)\) (approximate), and the x - intercepts are at \((-6,0)\) and \((-2,0)\). So two points on the axis of symmetry (the vertical line \(x=-4\)) are \((-4,0)\) and \((-4,1)\). But if we consider the two symmetric points on the parabola (not on the axis), they are \((-6,0)\) and \((-2,0)\), and the axis of symmetry is the line \(x=-4\), so two points on the axis are \((-4,0)\) and \((-4,1)\) (or any two points with \(x=-4\)).

Answer:

Two points: \((-4,0)\) and \((-4,1)\) (or other points with \(x = - 4\), e.g., \((-4,2)\) etc. But the most appropriate from the x - intercepts and vertex are \((-4,0)\) and \((-4,1)\))