Question

Write an equation g(x) for the parent function f(x)=x^2 re-elected in the x-axis, stretched by a factor of 7, shifted left 5 units, and shifted down 2 units

Answer 1

The resulting equation is
`g(x) = 7\cdot (x+5)^(2)-2`.

Explanation:

`g(x)` is obtained by making three operation on parent function
`f(x)`, whose procedured is presented below:

Streching

`f'(x) = k\cdot f(x)`, where
`k > 0`

Horizontal shift

`f''(x) = f'(x+r)`, where
`r > 0` when
`f'` is translated leftwards (+x direction), otherwise it is translated rightwards (-x direction).

Vertical shift

`g(x) = f''(x)+c`, where
`c > 0` when
`f''` is translated upwards (+y direction), otherwise it is translated downwards. (-y direction)

If
`f(x) = x^(2)`, then
`g(x)` is:

Stretching

`f'(x) = 7\cdot x^(2)`

Horizontal shift

`f''(x) = 7\cdot (x+5)^(2)`

Vertical shift

`g(x) = 7\cdot (x+5)^(2)-2`

The resulting equation is
`g(x) = 7\cdot (x+5)^(2)-2`.