Question

30 POINTS!

The graph of h is a translation 4 units down and 7 units right of the graph of f(x) = x^2 + 6x. For each value of x, g(x) is 80% of h(x). Write a rule for g.

Answer 1

The rule to transform
`f(x)` into
`g(x)` is:

`g(x) = 0.8\cdot [f(x-7) - 4]`

For
`f(x) = x^(2) + 6\cdot x`:
`g(x) = 0.8\cdot x^(2) -6.4\cdot x +2.4`

Explanation:

A vertical translation of a function is described by the following operation:

`y' = y + k` (1)

Where:

`y` - Original function.

`y'` - Translated function.

`k` - Vertical translation factor (
`k > 0` - Upwards)

And a horizontal translation of a function is described by the following operation:

`y' = f(x-k)` (2)

Where:

`x` - Independent variable.

`k` - Horizontal translation factor (
`k > 0` - Rightwards)

And the dilation of a function is defined by this operation:

`y' = k\cdot y` (3)

Where
`k` is the dilation factor (
`0 < k < 1` - Contraction)

The rule to transform
`f(x)` into
`g(x)` is:

`g(x) = 0.8\cdot [f(x-7) - 4]`

If we know that
`f(x) = x^(2) + 6\cdot x`, then
`g(x)` is:

`g(x) = 0.8\cdot [(x-7)^(2)+6\cdot (x-7) - 4]`

`g(x) = 0.8\cdot [(x^(2)-14\cdot x + 49)+(6\cdot x -42) - 4]`

`g(x) = 0.8\cdot (x^(2)-8\cdot x + 3)`

`g(x) = 0.8\cdot x^(2) -6.4\cdot x +2.4`