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Step 1:: Identify the shape of the graph
The given graph is a parabola opening upwards, which is the typical shape of a quadratic function.
Step 2:: Determine the general form of the quadratic function
The general form of a quadratic function is f(2$) = ax^2 + bx + c. In this case, since the parabola opens upwards, the coefficient 'a' must be positive.
Step 3:: Find the vertex of the parabola
The vertex of the parabola is (h, k), where h = -b/(2a) and k = f(h). From the graph, we can estimate the coordinates of the vertex to be around (1.5, 0).
Step 4:: Calculate the coefficients 'a', 'b', and 'c'
Using the vertex coordinates (1.5, 0), we can calculate 'a' and 'b' as follows: h = -b / (2a)
Step 5:5 = -b / (2a)
b = - 3 / 2 * a Since the parabola passes through the point (0, 2), we can find 'c' as follows: c = f(2$) = 2 Now we have: f(2$) = ax^2 - (3 / 2)ax + 2
Step 6:: Constrain 'a' using another point on the graph
The graph passes through the point (2, 4). We can use this information to find the exact value of 'a': f(2$) = a(2$)^2 - (3 / 2)a(2$) + 2 = 4 4a - 3a + 2 = 4 a = 2 So, the function is: f(2$) = 2x^2 - 3x + 2
Final Answer
The function shown in the graph is f(2$) = 2x^2 - 3x + 2.
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