SOLUTION: Find the vertex, axis of symmetry, maximum value or minimum value, x and y intercepts. f(x)=-3/11(x-7)2-9

Find the vertex, axis of symmetry, maximum value or minimum 
value, x and y intercepts. 

f(x) = -3/11(x - 7)² - 9

You must learn these facts about an equation of the form:

f(x) = a(x - h)² + k

1. The graph is shaped like this È if " a " is positive.
   The graph is shaped like this Ç if " a " is negative.

2. The vertex is the lowest point on the graph if it looks 
   like this È 
   The vertex is the highest point on the graph if it looks 
   like this Ç
   The vertex is the point (h, k)

3. The graph always goes through the points
   (h + 1, k + a) and (h - 1, k + a)

4. The axis of symmetry is the vertical line which bisects
   the parabola.  Its equation is x = h.

5. To find the y intercept, substitute x = 0 and solve for
   y. Then the y-intercept is the point (0, that value of y)
   There will always be a y-intercept.

6. 5. To find the x intercept(s), substitute y = 0 and solve for
   x. Then the x-intercept(s) are the point 
   (that or those values of x, 0)
   Sometimes there are no x-intercept(s).  This will be the case
   if the value of x is imaginary. 
   
Now using your equation:

f(x) = -3/11(x - 7)² - 9

compare that to 

f(x) = a(x - h)² + k

a = -3/11, h = 7, k = -9

1. The graph is shaped like this Ç because
   " a " is negative.

2. The vertex is the highest point on the graph. 
   The vertex is the point (h, k) = (7, -9)

3. The graph goes through the points
   (h - 1, k + a) and (h + 1, k + a), which are
   (7 - 1, -9 - 3/11) and (7 + 1, -9 - 3/11) or
   (6, -102/11) and (8, -102/11) or about
   (6, -9.3) and (8, -9.3)   

4. The axis of symmetry is the vertical line which bisects
   the parabola.  Its equation is x = h or x = 7.

5. To find the y-intercept, we substitute x = 0 and solve for
   f(0).

   f(x) = -3/11(x - 7)² - 9
   f(0) = -3/11(0 - 7)² - 9
   f(0) = -3/11(-7)² - 9
   f(0) = -3/11(49) - 9
   f(0) = -147/11 - 9
   f(0) = -246/11

   So the y-intercept (0, -246/11), which
   is about (0, -22.4) 

6. 5. To find the x intercept(s), we substitute f(x) = 0 
   and solve for x. Then the x-intercept(s) are the point 
   (that or those values of x, 0)

   f(x) = -3/11(x - 7)² - 9

      0 = -3/11(x - 7)² - 9
   
Multiply thru by 11

                    0 = -3(x - 7)² - 99

Divid through by -3

                    0 = (x - 7)² + 33
                    0 = (x - 7)(x - 7) + 33
                    0 = (x² - 14x + 49) + 33
                    0 = x² - 14x + 49 + 33
                    0 = x² - 14x + 82

We now find out if this has real or imaginary
solutions by using the discriminant B²-4AC where
A=4, B=-14, C=82

B²-4AC = (-14)²-4(4)(82) = -1116

When the discriminant is negative, the solutions
are imaginary, so there are no x-intercepts.

Here is the graph
 


Here it is with the line of symmetry x = 7



Edwin