We want to sketch the graph of a quadratic function. Let's think about what the given information tells us.
A quadratic function is a 2-degree function — this degree tells us that the function can have two or zero real zeros.
x-Intercepts at -4 and 2 — this tells us the location of all real zeros.
Range: y ≥ - 3 — this tells us the parabolaopens up and its vertex is the point (h,- 3).
Let's find the vertex of our function and then draw the parabola through the given points.
Vertex
We know that the vertex of the quadratic function lies on its axis of symmetry. The axis of symmetry is halfway between (p,0) and (q,0). Since we know that the roots of our function are - 4 and 2, the axis of symmetry of our parabola is halfway between (- 4,0) and (2,0).
x=p+q/2 ⇒ x=- 4+ 2/2=- 2/2=- 1
We found that the axis of symmetry is the vertical line x=- 1 and since the vertex lies on the axis of symmetry, its x-coordinate is - 1. Therefore vertex is the point (- 1, - 3).
Graph
Knowing the roots of the function and its vertex, we can already draw the graph through the previously found points.
