Quadratic Function Equation Determination
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.
