Question

The table represents a quadratic function. Write an equation of the function in standard form. x,4,5,6,7 f(x),3,2,3,6 The function is f(x)

Answer 1

The equation of the function in standard form is:

f(x) =
`x^2` - 10x + 27

The plot suggests a quadratic relationship between x and f(x). To find the equation of the function in standard form, which is f(x) = a
`x^2` + bx + c, we need to determine the coefficients a, b, and c using the given points.

Given the symmetry of the points around x = 5.5, it seems that this is the axis of symmetry for the parabola, which suggests that the vertex form of the equation might be useful for determining the coefficients. The vertex form of a quadratic function is f(x) = a
`(x - h)^2` + k, where (h, k) is the vertex of the parabola.

Since the function has a minimum value at x = 5, this is likely the h value in the vertex form. The corresponding f(x) value at this point is 2, which is the k value. Thus, the vertex is (5, 2).

Now, we can use the vertex form to find a by plugging in one of the other points, for example (4, 3):

3 = a
`(4 - 5)^2` + 2

Solving for a gives us:

3 =
`a(1)^2` + 2

1 = a

So, a = 1. Now we have the vertex form of the equation:

f(x) =
`(x - 5)^2` + 2

To convert this to standard form, we expand the squared term:

f(x) = (
`x^2` - 10x + 25) + 2

f(x) =
`x^2` - 10x + 27

Therefore, c