SOLUTION: PLEASE HELP ME SOLVE THIS: Determine whether the quadratic function f(x)=3x^+18x-23 has a maximun value or minimum value. Find the range of the function f(x)expressed using int

Algebra ->  Functions -> SOLUTION: PLEASE HELP ME SOLVE THIS: Determine whether the quadratic function f(x)=3x^+18x-23 has a maximun value or minimum value. Find the range of the function f(x)expressed using int      Log On


   

Question 243399: PLEASE HELP ME SOLVE THIS:
Determine whether the quadratic function f(x)=3x^+18x-23 has a maximun value or minimum value. Find the range of the function f(x)expressed using interval notation.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
equation is:

3x^2 + 18x - 23

It doesn't look it will factor easily, so use the quadratic formula.

quadratic equation general form is ax^2 + bx + c

In this equation:

a = 3
b = 18
c = -23

quadratic formula is %28%28-b%29+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29

b^2 - 4ac is the discriminant.

formula becomes:

%28%28-18%29+%2B-+sqrt%2818%5E2-4%2A3%2A%28-23%29%29%29%2F%282a%29

this becomes:

{{((-18) +- sqrt(600))/(6)}}} which becomes:

x = 1.082482905 or x = -7.082482905

The roots are real and those are the points where the equation crosses the x-axis.

The minimum / maximum point of this equation is given by the formula:

x = -b/2a and y = f(-b/2a)

the value for x = -(18) / (2*3) = -18/6 = -3

the value for y = f(-b/2a) = f(-3) = 3*(-3)^2 + 18*(-3) - 23 = 27 - 54 - 23 = 27 - 77 = -50

The max / min point is equal to (-2,-50)

The range of this function is dependent on the domain.

The domain of this function looks like it is all real values of x because x can be positive and negative and is all real (no negative square roots and no divisions by 0 to restrict the domain).

The range is all real values of y but the minimum / maximum value of y is determined by the minimimum / maximum point.

In this case the min / max point is a minimum because the x^2 term is positive.

This means the range of the function will be all real values of y greater than or equal to the minimum point which is -50.

In interval notation this would be x >= -50

It can also be written as -50 <= x < infinity which in symbol form looks like this -50+%3C=+x+%3C+infinity

a graph of your equation looks like this:

graph%28600%2C600%2C-10%2C10%2C-100%2C100%2C3x%5E2%2B18x-23%2C-50%29

I placed a horizontal line at y = -50 to show you that the minimum point was there.

Since I'm not sure which form of interval notation you are looking for, the only other form interval notation I know of would be:

y = [ -50, infinity)

This means the value of y is greater than or equal to 50 and smaller than infinity.