Question

Which function has a minimum and is transformed to the right and down from the parent function, f(x) = x2?

g(x) = –9(x2 + 2x + 1) – 7
g(x) = 4(x2 – 6x + 9) + 1
g(x) = –3(x2 – 8x + 16) – 6
g(x) = 8(x2 – 6x + 9) – 5

Answer 1

The last choice is the one you want

Explanation:

First of all, a parabola has a minimum value if it is a positive parabola, one that opens upward. The first and the third parabolas are negative so they open upside down. That leaves us with choices 2 and 4. We find the side to side and up or down movement by finishing the completion of the square that has already been started for us. Do this by factoring what's inside the parenthesis into a perfect square binomial.

The second one factored becomes:

`g(x)=4(x-3)^2+1`

which reflects a shift to the right 3 and up 1. Not what we are asked to find.

The fourth one factored becomes:

`g(x)=8(x-3)^2-5`

which reflects a shift to the right 3 and down 5. That's what we want!

Answer 2

g(x) = 8(x2 – 6x + 9) – 5

Explanation:

A function with a form (x - a)^2 - b where a and b are some constants are a transformation to the right and down from the parent function, f(x) = x^2. Then, option:

g(x) = 4(x2 – 6x + 9) + 1 = 4(x - 3)^2 + 1

is discarded

The vertex form of a quadratic equation is f(x) = c*(x - h)^2 + k, where c, h and k are all constant, and point (h,k) is the vertex of the quadratic function. If c > 0, then the vertex is a minimum and if c < 0 then the vertex is a maximum. Therefore, options:

g(x) = –9(x2 + 2x + 1) – 7 = -9(x + 1)^2 - 7

g(x) = –3(x2 – 8x + 16) – 6 = -3(x - 4)^2 - 6

are dicarded

In consequence, the correct option is:

g(x) = 8(x2 – 6x + 9) – 5 = 8(x - 3)^2 - 5