To find the quadratic function that represents the parabola with a given vertex and y-intercept, we can use the vertex form of a quadratic equation, which is:
f(x) = a(x – h)² + k
In this equation, (h, k) is the vertex of the parabola. Here, the vertex given is (2, 20), so we have:
f(x) = a(x – 2)² + 20
Next, we need to find the value of ‘a’. To do this, we can use the y-intercept that is provided, which is the point (0, 12). This means that when x = 0, f(x) = 12. We can substitute these values into the equation:
12 = a(0 – 2)² + 20
Solving for ‘a’:
12 = a(4) + 20
12 – 20 = 4a
-8 = 4a
a = -2
Now that we have the value of ‘a’, we can write the complete equation:
f(x) = -2(x – 2)² + 20
To express this in standard form (ax² + bx + c), we need to expand it:
f(x) = -2(x² – 4x + 4) + 20
f(x) = -2x² + 8x – 8 + 20
f(x) = -2x² + 8x + 12
Thus, the quadratic function that represents the parabola in standard form is:
f(x) = -2x² + 8x + 12