2.2 Linear transformation

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)=|x|[/math] to [math]g(x)=2|-x-2|-2[/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)_.

Describe linear transformations from the parent function [math]f(x)=x^3[/math] to [math]g(x)=-(4(x-3))^3+2[/math].

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

Let [math]f(x)[/math] be a function. Which of the following function(s) having the same [math]x[/math]-intercepts as [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(t)=1000-997\cdot(0.8)^t[/math] be a function representing the height of the airplane where [math]t[/math] is measured in seconds and [math]f(t)[/math] is measured in meters. Suppose we want to change the unit of height from meters to kilometers and the unit of time from seconds to hours. What is the new function?

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)=\dfrac{1}{x^2+1}[/math]. We stretch [math]f(x)[/math] by the factor 2 horizontally and vertically, and translate 4 units left and 1 unit up. What is the new function?

Let [math]f(x)=2x^3-6[/math]. We reflect [math]f(x)[/math] about the [math]y[/math]-axis, compress by the factor 3 vertically, and translate 4 units right. What is the new function?

The graph of function [math]f(x)=\sqrt{1^2-x^2}[/math] is an upper semicircle with center [math](0,0)[/math] and radius 1. If the graph of [math]g(x)=3f\left(\dfrac{1}{3}x\right)-3[/math] is also an upper semicircle, what is the center and radius of [math]g(x)[/math]?

Consider a linear function [math]f(x)=-\dfrac{1}{4}x-3[/math] where [math]m[/math] is negative. If the graph of [math]g(x)=f(x-k)[/math] does not pass the third quadrant where [math]k[/math] is an integer, what is the minimum of [math]k[/math]?

The table shows some values of function [math]f(x)[/math]. If [math]g(x)=-f(x-2)+4[/math], which of the following values is largest?

Let [math]f(x)[/math] be a function having two [math]x[/math]-intercepts [math]3[/math] and [math]-2[/math]. If [math]g(x)=-2f\left(\dfrac{1}{3}x+3\right)[/math]. What is the summation of the [math]x[/math]-intercepts of [math]g(x)[/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?

Let [math]f(x)[/math] be a nonlinear function passing through [math](0,-3)[/math] and [math](4,0)[/math]. If [math]g(x)=3f(-2x)[/math], what is the summation of the [math]x[/math]-intercept and [math]y[/math]-intercept of [math]g(x)[/math]?

Let [math]f(x)[/math] and [math]g(x)[/math] be two functions. If [math]f(2x+4)=-3g(x)-4[/math]. Describe the linear transformation from [math]f(x)[/math] to [math]g(x)[/math].

Let [math]f(x)[/math] be a function. If [math]f(2x)=2f(x)[/math], which of the following function(s) could be [math]f(x)[/math]?

The domain of function [math]f[/math] is all numbers between [math]-4[/math] and [math]6[/math] inclusive, and the range is all numbers between [math]-3[/math] and [math]5[/math], inclusive. What are the domain and range of [math]g(x)=3f(\dfrac{x}{2})[/math]?

The domain of [math]f(x)[/math] is [math][-4,6][/math], and the range is [math][-3,5][/math]. Given [math]g(x)=3f(x+1)+5[/math], what are the domain and range of [math]g(x)[/math]?

If [math]f(x)=x^2-2x+3[/math], which of the following is equal to [math]g(x)=f(x+1)[/math]?

The figure below is [math]f(x)[/math], which of the following is the graph of [math]y=k-f(x)[/math], where [math]k[/math] is a positive number?

In the figure above, a portion of the graph of [math]y=f(x)[/math] is shown. What are the coordinates of the [math]y[/math]-intercept of the graph of [math]y=f(x+2)[/math]?

(Calculator) If [math]f(x)=x^4-4x+1[/math] and [math]g(x)=f(x+2)[/math], which of the following statements are true?
I. [math]f[/math] and [math]g[/math] have one common zero.
II. The graphs of [math]f[/math] and [math]g[/math] have one intersection point.
III. [math]f[/math] and [math]g[/math] have the same range.