Question

G(x)=-2(x-4)^2+1, please describe ALL transformations

Answer 1

1. Translation of 4 units right.

2. Vertical stretch by a factor of 2:

3. Reflection in the x-axis:

4. Translation of 1 unit up.

Explanation:

Transformations

`\textsf{For $a > 0$}:`

`f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.`

`a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis (vertically) by a factor of $a$}.`

`-f(x) \implies f(x) \: \textsf{reflected in the $x$-axis}.`

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}.`

Given function:

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

The parent function of the given function is:

`f(x)=x^2`

When determining the sequence of transformations when the function contains more than one transformation, follow the order of operations (PEMDAS).

1. Translation of 4 units right.

`f(x-4) \implies g(x)=(x-4)^2`

2. Vertical stretch by a factor of 2:

`2f(x-4)\implies g(x)=2(x-4)^2`

3. Reflection in the x-axis:

`-2f(x-4)\implies g(x)=-2(x-4)^2`

4. Translation of 1 unit up.

`-2f(x-4)+1\implies g(x)=-2(x-4)^2+1`