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

Answer:

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

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

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

Step-by-step 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:

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

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

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

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