To determine the quadratic function that represents a parabola with a given vertex and y-intercept, we can start with the vertex form of a quadratic function, expressed as:
y = a(x – h)² + k
where:
- (h, k) is the vertex of the parabola.
- a is a coefficient that affects the width and direction of the parabola.
In this case, the vertex is given as (2, 20), so we can set h = 2 and k = 20. This gives us the equation:
y = a(x – 2)² + 20
Next, we can use the y-intercept to find the value of a. The y-intercept occurs when x = 0, and we are given that the y-intercept is (0, 12). Substituting these values into the equation:
12 = a(0 – 2)² + 20
Now, solve for a:
- 12 = a(4) + 20
- 12 – 20 = 4a
- -8 = 4a
- a = -2
Now we have the value of a, and we can substitute it back into the vertex form equation:
y = -2(x – 2)² + 20
To convert this into standard form, we expand the equation:
y = -2(x² – 4x + 4) + 20
Distributing the -2, we have:
y = -2x² + 8x – 8 + 20
Combining like terms:
y = -2x² + 8x + 12
Therefore, the quadratic function representing the parabola in standard form is:
y = -2x² + 8x + 12