How Do You Determine the Minimum Value of a Quadratic Function?

[Ewan_C]

[SOLVED] Solving quadratic

Homework Statement

1a) Solve f(x) = x^2- 4x+m in the form f(x) = (x-a)^2+ b
1b) What is the smallest value f(x) can have?

The Attempt at a Solution

1a) Seems simple enough. I set f(x) to 0 and used the completing the square method to solve. Ended up with f(x)=(x-2)^2+ m-4.

I don’t know how to approach 1b) though. I’m assuming I’ve made a mistake somewhere – there is no smallest value f(x) can have without knowing the value of m. Is there a way to find m that I haven't picked up on, or would the answer just be m - 4?

Thanks for any advice

 

[sutupidmath]

Homework Statement

1a) Solve f(x) = x^2- 4x+m in the form f(x) = (x-a)^2+ b
1b) What is the smallest value f(x) can have?

The Attempt at a Solution

1a) Seems simple enough. I set f(x) to 0 and used the completing the square method to solve. Ended up with f(x)=(x-2)^2+ m-4.

I don’t know how to approach 1b) though. I’m assuming I’ve made a mistake somewhere – there is no smallest value f(x) can have without knowing the value of m. Is there a way to find m that I haven't picked up on, or would the answer just be m - 4?

Thanks for any advice

Looks fine to me.

 

[HallsofIvy]

Homework Statement

1a) Solve f(x) = x^2- 4x+m in the form f(x) = (x-a)^2+ b
1b) What is the smallest value f(x) can have?

The Attempt at a Solution

1a) Seems simple enough. I set f(x) to 0 and used the completing the square method to solve. Ended up with f(x)=(x-2)^2+ m-4.

I don’t know how to approach 1b) though. I’m assuming I’ve made a mistake somewhere – there is no smallest value f(x) can have without knowing the value of m. Is there a way to find m that I haven't picked up on, or would the answer just be m - 4?

Thanks for any advice

No, what you have given is fine. No matter what x is (x- 2)² can never be lower than 0 so f(x)= (x-2)² + m- 4 can never be lower than m- 4. Since you are not given a specific value of m, that is all you can do.

 

[Ewan_C]

Okay, thanks for that