Solving Quadratic Equations and Finding Equation in Standard Form
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 - 9
6 = a(3)^2 - 9
6 = 9a - 9
Now, add 9 to both sides to get:
6 + 9 = 9a
15 = 9a
Divide both sides by 9 to solve for 'a':
15 / 9 = a
a = 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) - 9
f(x) = (5/3)x^2 - 10x + 15 - 9
f(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.
