The graph of a quadratic function

Question

The graph of a quadratic function g(x) has vertex (4,2). It is known that g(6) = 22.

a. Write the formula for g(x) in standard/vertex form g(x) = A(x-h)² + K. Hint: you should be able to easily find the values of h,k. Then all you have to do is figure out what A is.

b. Use algebra to simplify your answer to the previous problem to write the formula g(x) in the form g(x)= Ax² + Bx + C.

Answer 1

(a) g(x) = A(x - h)² + k

 

From the vertex (4, 2), 2 = A(4 - h)² + k.

In this form, the vertex value of y has only one solution.  All other values of y have either 0 or 2 values.

For the vertex, this occurs when h = 4.  This means that k = 2

 

Then g(x) = A(x - 4)² + 2

 

Also, g(2) = 22 = A(6 - 4)² + 2 = 4A + 2

This means A = 5

 

g(x) = 5(x - 4)² + 2

 

(b) g(x) = 5(x² - 8x + 16) + 2

             = 5x² - 40x + 82