Question - Solving Quadratic Equations and Finding Equation in Standard Form
Solution:
To find the equation of a quadratic function in the form of f(x) = ax^2 + bx + c, we should identify key points on the graph to help us determine the values of a, b, and c. From the graph provided, there are two points we can use:1. The vertex of the parabola, which is (3, -9).2. Another point on the curve, which is (6, 6).The vertex form of a quadratic equation is given by:f(x) = a(x - h)^2 + k,where (h, k) is the vertex of the parabola.Since we know the vertex is (3, -9), the vertex form of our equation becomes:f(x) = a(x - 3)^2 - 9.We can use the other given point (6, 6) to find the value of 'a'. Plugging the point into the vertex form gives us:6 = a(6 - 3)^2 - 96 = a(3)^2 - 96 = 9a - 9Now, add 9 to both sides to get:6 + 9 = 9a15 = 9aDivide both sides by 9 to solve for 'a':15 / 9 = aa = 5/3.Now we can write the complete equation for the parabola:f(x) = (5/3)(x - 3)^2 - 9.To write it in standard form, we can expand the squared term and distribute 'a':f(x) = (5/3)(x^2 - 6x + 9) - 9f(x) = (5/3)x^2 - 10x + 15 - 9f(x) = (5/3)x^2 - 10x + 6.So the equation of the quadratic function in standard form is:f(x) = (5/3)x^2 - 10x + 6.
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