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Explanation:
Compare the given equation to the vertex form y = a(x-h)^2 + k
You'll see that a = 3, h = 4 and k = 7
The vertex is (h,k) = (4,7)
The value of 'a' determines if the parabola opens upward or downward. Since a > 0, it opens upward. This furthermore leads to the vertex being the lowest point on the parabola.
The value of 'a' also determines how vertically stretched or compressed the parabola is.
Answer:
1. The vertex of the graph of the function is (-4, 7)
2. The vertex represents a minimum value
3. The equation represented by the new graph is g(x) = (x + 4)² + 1
Step-by-step explanation:
The given quadratic function is given in vertex form, y = a·(x - h)² + k, as follows;
g(x) = (x + 4)² + 7
1. By comparing the given quadratic function and the vertex form of a quadratic equation, we have;
a = 1, h = -4, and k = 7
The vertex of the graph of the function, (h, k) = (-4, 7)
2. Given that a = 1 > 0, the graph of the quadratic function opens upwards and the vertex represents a minimum value
3. Shifting the graph 6 units down from where it is now will give;
The vertex = (h, k - 6) = (-4, 7 - 6) = (-4, 1)
h = -b/(2·a), k = (-b²/(4·a) + c = -a·h² + c
c = k + a·h²
Therefore, initial value of the constant term, c = -1×(-4)² + 7 = 23
After the shifting the graph 6 units down, c = 23 - 6 = 17
∴ k = 1 = -a·(-4)² + 17
∴ -16/16 = -1 = -a
a = 1
Therefore, the equation represented by the new graph is g(x) = (x + 4)² + 1.
