Answer:
y = - ½(x - 2)² + 3
Step-by-step explanation:
Given the vetex, (2, 3), as the maximum point on the graph, along with the y-intercept, (0, 1):
We can substitute these values into the given vertex form of the quadratic equation:
y = a(x - h)² + k
where:
- a = determines whether the graph opens up or down, and makes the parent function wider or narrower. If a is positive, the graph opens up; if a is negative, the graph opens down. If 0 < a < 1, the graph is wider than the parent function. If a > 1, the graph is narrower than the parent function.
- (h, k ) = vertex
- h = determines how far left or right the parent function is translated.
- k = determines how far up or down the parent function is translated.
Use vertex = (2, 3) and y-intercept (0, 1) to solve for "a":
y = a(x - h)² + k
1 = a(0 - 2)² + 3
1 = a(-2)² + 3
1 = a(4) + 3
Subtract 3 from both sides:
1 - 3 = a(4) + 3 - 3
-2 = 4a
Divide both sides by 4 to solve for a:
-2/4 = 4a/4
- ½ = a
Therefore, the value of a = - ½. This means that the graph opens downward, and that the parabola is wider than the parent function.
The quadratic equation in vertex form given a = - ½, and vertex (2, 3) is:
y = - ½(x - 2)² + 3
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