# Math Help - Given the absolute minimum determine the value of a function

1. ## Given the absolute minimum determine the value of a function

The graph of a function defined on [-3,3] by
f(x)= x^2 + ax + b
and has an absolute minimum at (-1,-3). Determine the value of f(1).

I'm having trouble figuring this problem out. I know the graph is a parabola. Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
Thank you!

2. Originally Posted by yzobel
The graph of a function defined on [-3,3] by
f(x)= x^2 + ax + b
and has an absolute minimum at (-1,-3). Determine the value of f(1).

I'm having trouble figuring this problem out. I know the graph is a parabola. <<<< OK
Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
Thank you!
1. The coefficient of x² is (+1): Therefore the parabola opens up and the absolute Minimum is at the vertex of the parabola.

2. Since the absolute minimum is (-3) at x = -1 use the vertex-form of the parabola:

$f(x) = a(x-k)^2+h$ where a determines the shape of the parabola and (k, h) is the vertex of the parabola.

3. You should come out with:

$f(x)=1 \cdot (x+1)^2-3~\implies~f(x)=x^2+2x-2$

4. Draw a sketch of the parabola.

3. Originally Posted by yzobel
The graph of a function defined on [-3,3] by
f(x)= x^2 + ax + b
and has an absolute minimum at (-1,-3). Determine the value of f(1).

I'm having trouble figuring this problem out. I know the graph is a parabola. Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
Thank you!
No, the constant "b" is merely f(0).

You have a calculus and non-calculus way to solve this.

Calculus: The derivative is the slope of the tangent, which is zero at the minimum.

Then using the x co-ordinate of the minimum, "a" can be found.
"b" can be found from the y co-ordinate of the minimum.

Non-calculus: Use the fact that the graph is symmetrical about the minimum, the line x=-1.

$f(1)=f(-3)\Rightarrow\ 1+a+b=9-3a+b\Rightarrow\ 4a=8\Rightarrow\ a=2$

Then you can get "b" from f(-1)=-3.

4. Thank so much guys!