Question

Which function has the least minimum value and what are its coordinates?

Answer 1

Function 1:

The quadratic function is given as:

`f(x)=2x^2-8x+1`

Factor out 2:

`f(x)=2(x^2-4x)+1`

Rewrite the expression in the bracket, as follows:

`f(x)=2(x^2-4x+4-4)+1`

This gives:

`\begin{gathered} f(x)=2(x^2-4x+4)-8+1 \\ f(x)=2(x^2-4x+4)-7 \end{gathered}`

Express as a perfect square:

`f(x)=2(x-2)^2-7`

A quadratic function is represented as:

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

Where: vertex (h,k)

By comparison, we have:

Vertex = (2, -7)

This represents the minimum of function 1.

Function 2:

From the graph, the minimum of function 2 is:

Vertex = (-1, -3)

(2,-7) is lesser than (-1,-3)

Answer: Function 1 has the least minimum value at a coordinate of (2,-7)