Answer:
Option 1: y = (x - 2)² + 2
Step-by-step explanation:
Given the vertex, (2, 2), of an upward-facing parabola, and using the y-intercept of the graph, (0, 6):
Definitions:
The vertex form of the quadratic equations is:
y = a(x - h)² + k
where:
(h, k) is the vertex.
- The value of h represents the horizontal translation of the graph. h > 0 represents the horizontal shift of the graph h units to the right. h < 0 shifts the graph |h | units to the left.
- The value of k represents the vertical translation of the graph. k > 0 shifts the graph k units upward. k < 0 shifts the graph |k | units down.
The sign of a determines the direction of the graph's opening. The value of a also determines the vertical stretch or shrink of the graph.
- If a is positive (or a > 1), then the graph opens upward. The value of a > 1 also represents the vertical stretch of the graph.
- If a is negative (or a < 0), then the graph opens down.
- The value of 0 < a < 1 represents the vertical shrink or compression of the graph.
Solution:
Substitute the values of the vertex (2, 2) and the y-intercept, (0, 6) into the vertex form to solve for the value of the coefficient, a:
y = a(x - h)² + k
6 = a(0 - 2)² + 2
6 = a(-2)² + 2
6 = 4a + 2
Subtract 2 from both sides
6 - 2 = 4a + 2 - 2
4 = 4a
Divide both sides by 4 to solve for a :
[tex]\displaytext\mathsf{\frac{4}{4} \:=\frac{4a}{4}}[/tex]
a = 1
Quadratic Equation in Vertex Form:
Therefore, given the vertex, (2, 2) and a = 1, the quadratic equation in vertex form is: y = (x - 2)² + 2, thereby matching Option 1.
