Answer:
Vertex form: y = 3(x + 2)² + 2
Step-by-step explanation:
Given the graph of an upward-facing parabola, whose vertex occurs at point (-2, 2):
Definition of terms:
The vertex form of a quadratic equation is given by: y = a(x - h)² + k
where:
(h, k) ⇒ vertex
h = indicates a horizontal translation of the graph.
k = indicates a vertical translation of the parabola.
x = h ⇒ axis of symmetry: the imaginary vertical line that divides the graph into two symmetrical parts.
a = direction of the parabola's opening; determines the wideness of the graph.
- a < 0: graph opens down.
- |a| > 1: graph is narrower than the parent graph, y = ax².
- |a| < 1: graph is wider than the parent graph.
Solution:
Now that we have established the necessary terms related to understanding the concept of quadratic equations, we can proceed with determining the equation of the graph.
Besides the vertex, (-2, 2), we must choose another point on the graph to solve for the value of a. Using the point from the graph, (-1, 5), along with the vertex, substitute these values into the vertex form to solve for a:
y = a(x - h)² + k
5 = a[-1 - (-2)]² + 2
5 = a(-1 + 2)² + 2
5 = a(1)² + 2
5 = a + 2
Subtract 2 from both sides:
5 - 2 = a + 2 - 2
3 = a
Therefore, the vertex form of the given graph of a parabola is:
y = 3(x + 2)² + 2
