Question - Quadratic Function Equation Determination
Solution:
The graph in the image depicts a quadratic function, which can be represented by the general equation:\[ g(x) = a(x - h)^2 + k \]where $$(h, k)$$ is the vertex of the parabola, and $$a$$ is a coefficient that determines the direction and width of the parabola. If $$a$$ is positive, the parabola opens upwards; if $$a$$ is negative, it opens downwards.In this graph, the vertex of the parabola is clearly at the point $$(2, 7)$$. Therefore, $$h = 2$$ and $$k = 7$$. Now, we can substitute these values into the equation:\[ g(x) = a(x - 2)^2 + 7 \]We have another point on the graph, $$(3, 4)$$. We can use this point to solve for the coefficient $$a$$. Substituting $$x = 3$$ and $$g(x) = 4$$ into the equation above, we get:\[ 4 = a(3 - 2)^2 + 7 \]\[ 4 = a(1)^2 + 7 \]\[ 4 = a + 7 \]\[ a = 4 - 7 \]\[ a = -3 \]Now that we have the value of $$a$$, we can write the final equation of the quadratic function:\[ g(x) = -3(x - 2)^2 + 7 \]This is the required equation of the quadratic function.
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