If the quadratic function is f(x) =x^2-9x+20. Derive the shape Vertex formula?

To find the vertex of the quadratic function f(x) = x^2 - 9x + 20, we can use the formula for the x-coordinate of the vertex, which is given by:

x = -b/2a

In this case, a = 1 and b = -9, so we have:

x = -(-9) / 2(1)
x = 9 / 2
x = 4.5

To find the y-coordinate of the vertex, we can substitute x = 4.5 back into the original function:

f(4.5) = (4.5)^2 - 9(4.5) + 20
f(4.5) = 20.25 - 40.5 + 20
f(4.5) = 0.75

Therefore, the vertex of the quadratic function f(x) = x^2 - 9x + 20 is (4.5, 0.75).

Shape of a Quadratic Function:

The graph of a quadratic function f(x) = ax^2 + bx + c is a parabola. The shape of the parabola depends on the value of the coefficient "a".

Vertex Formula:

The vertex of a parabola is the point where the function reaches a minimum or maximum value. The vertex of the parabola for the function f(x) = ax^2 + bx + c is given by:

```
Vertex = (-b/2a, f(-b/2a))
```

Derivation of the Vertex Formula for f(x) = x^2 - 9x + 20:

1. Rewrite the function in the standard form: f(x) = x^2 - 9x + 20 = x^2 - 9x + (4 + 16) = x^2 - 9x + 4 + 16

2. Identify the coefficients: a = 1, b = -9, c = 20

3. Substitute the coefficients into the vertex formula: Vertex = (-b/2a, f(-b/2a)) = (-(-9)/2(1), f(-(-9)/2(1)))

4. Simplify the expression: Vertex = (9/2, f(9/2))

5. Find the y-coordinate of the vertex by evaluating f(x) at x = 9/2: f(9/2) = (9/2)^2 - 9(9/2) + 20 = 81/4 - 81/2 + 20 = 11/4

Therefore, the vertex of the parabola for the function f(x) = x^2 - 9x + 20 is (9/2, 11/4).