Write the equation of the quadratic function with roots -2 and 4 and a vertex at (1, 9).

since roots are -2 and 4,

equation must be

y = a(x+2)(x-4)
but (1,9) lies on it, so
9 = a(3)(-3)
a = -1

y = -1(x+2)(x-4)
or
y = -x^2 + 2x + 8
or
y = -(x-1)^2 + 9

To find the equation of a quadratic function with roots -2 and 4 and a vertex at (1, 9), you can use the factored form of a quadratic equation, which is:

f(x) = a(x - r1)(x - r2)

where r1 and r2 are the roots of the quadratic function, and a is a constant.

In this case, the roots are -2 and 4, so we have:

f(x) = a(x - (-2))(x - 4)

Since the vertex is at (1, 9), we can also use this information to find the value of a. The x-coordinate of the vertex, 1, is given by the formula:

x = -b / (2a)

Substituting the x-coordinate and y-coordinate of the vertex, we have:

1 = -(-2 + 4) / (2a)
1 = 2 / (2a)
1 = 1 / a

Solving for a, we find that a = 1.

Now we can substitute this value of a back into the equation:

f(x) = (x - (-2))(x - 4)
f(x) = (x + 2)(x - 4)

Expanding and simplifying:

f(x) = x^2 - 2x - 8

So, the equation of the quadratic function is f(x) = x^2 - 2x - 8.

To write the equation of a quadratic function, we need to use the standard form:

f(x) = a(x - h)^2 + k,

where (h, k) represents the coordinates of the vertex.

Given that the roots are -2 and 4, we know that the quadratic function can be written in factored form as:

f(x) = a(x - (-2))(x - 4).

To find the value of 'a', we can substitute the coordinates of the vertex, (1, 9), into the equation:

9 = a(1 - (-2))(1 - 4).

Simplifying the right side:

9 = a(3)(-3).

Solving for 'a':

9 = -9a.

Now, divide both sides by -9:

a = -1.

Therefore, the equation of the quadratic function is:

f(x) = -1(x + 2)(x - 4).

Expanding this equation:

f(x) = -1(x^2 - 2x - 4x + 8).

Simplifying further:

f(x) = -x^2 + 6x - 8.

Hence, the equation of the quadratic function is f(x) = -x^2 + 6x - 8.