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.

Question

asked May 27, 2022 30.5k views

1 Answer

PatrickNLT · Answer 1 · 2022-06-02T13:06:51+0000

Answer:

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:

- Original function.

- Translated function.

- 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:

- Independent variable.

- Horizontal translation factor (
k > 0 - Rightwards)

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

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

Where
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:

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

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

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

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