2.2 Linear transformation (in class)

Describe linear transformations from the parent function [math]f(x)=x^2[/math] to [math]g(x)=-3(3x-3)^2+3[/math].

Describe linear transformations from the parent function [math]f(x)=\sqrt{x}[/math] to [math]g(x)=-\sqrt{\dfrac{-x}{2}+2}[/math].

First, we reflect the [math]x[/math]-axis and [math]y[/math]-axis. Next, we horizontally _(a)_ by the factor [math]2[/math]. Lastly, we translate _(b)_ units(s) _(c)_.

Let [math]f(x)[/math] be a function. Which function’s domain is different from [math]f(x)[/math]?

Consider a function [math]f(x)[/math] with domain [math][-1,4][/math] and range [math][0,6][/math]. Let [math]g(x)=-f(2x+4)-5[/math].

(a) If the domain of [math]g(x)[/math] is [math][a,b][/math], what is the value of [math]b[/math]?

(b) If the range of [math]g(x)[/math] is [math][c,d][/math], what is the value of [math]c[/math]?

Let [math]f(x)[/math] be a linear function with slope [math]a[/math] and [math]y[/math]-intercept [math]b[/math]. We stretch [math]f(x)[/math] by the factor 2 horizontally and vertically, and translate 4 units left and 1 unit up. The new function is equivelant to [math]f(x)[/math]. Find the value of [math]4a+b[/math].

Let [math]f(x)[/math] and [math]g(x)[/math] be two functions. If [math]f(x-2)=g(x)-2[/math], which of the following statement(s) is/are true?

The figure shows a graph of the function [math]f(x)[/math].

(a) If the domain of [math]f(x)[/math] is [math][a,b][/math], what is [math]a+b[/math]?

(b) If the range of [math]f(x)[/math] is [math](c,d][/math], what is [math]c+d[/math]?

(c) Evaluate [math]f(f(3))[/math].

(d) Since [math]f(x)[/math] does not satisfy the _(horizontal/vertical)_ line test, [math]f(x)[/math] is not a one to one function.

(e) Let [math]g(x)=2f(-2x+1)+3[/math]. Find the maximum of [math]g(x)[/math].

Mr. Donkey solves the linear transformations from [math]f(x)=\dfrac{1}{1+x^2}[/math] to [math]g(x)=-\dfrac{3}{1+2(x+3)^2}-1[/math]. His work is shown below:
[math]\bullet[/math] Step 1: Reflect about the [math]x[/math]-axis
[math]\bullet[/math] Step 2: Vertically stretch by the factor 3
[math]\bullet[/math] Step 3: Horizontally compress by the factor 2
[math]\bullet[/math] Step 4: Translate 3 units left and 1 unit down
Which step is incorrect?