Transformations Of All Functions

Reflection 2 would go here

Graphical transformations by comparing two functions and listing the graphical transformations

Source: A problem from my notes

Explanation:

In this artifact, I compare f(x) and g(x), and then list the graphical transformations required to get from f(x) to g(x).

I got the solution by referencing the formula: \(a*f(b(x-c))+d\)

\(g(x)\) fits the formula like so: \(3 * f(1(x-1)) + 2\)

Artifact:

Describe how the graph of \(f(x) = \sqrt{x}\) can be transformed into \(g(x) = 3 * \sqrt{(x - 1)} + 2\)

  1. Horizontal shift of f(x) by one unit beacuse c shifts the graph horizontally by d units.
  2. Vertical shift of f(x) by two units because d shifts the graph vertically by d units.
  3. Vertical stretch by magnitude of three because \(a > 1\)

Graphical transformations by rewriting a function from a list of transformations

Source: From my notes

Explanation:

This artifact demonstrates graphical transformations by rewriting a function from a list of transformations.

Like the previous proficiency, I got the solution by referencing the formula: \(a*f(b(x-c))+d\)

Here are the steps I took to get from \(\sqrt{x}\) to \(-4 * \sqrt{3(x-2)} + 5\), in order.

  1. \(-\sqrt{x}\)
  2. \(-4\sqrt{x}\)
  3. \(-4\sqrt{3x}\)
  4. \(-4\sqrt{3(x-2)}\)
  5. \(-4\sqrt{3(x-2)+5}\)

Artifact:

Transform \(f(x) = \sqrt{x}\) into \(g(x)\)

  1. Reflect over the x-axis
  2. Vertical stretch by a magnitude of four
  3. Horizontal shrink by a magnitude of \({1 \over 3}\)
  4. Horizontal shift by two units.
  5. Vertical shift by five units.
\[g(x) = -4 * \sqrt{3(x-2)} + 5\]

Graphical transformations by transforming a graph given transformations

Source: Made it up

Explanation:

This artifact demonstrates graphical transformations by transforming a graph given transformations.

Like the previous proficiencies, I got the solution by referencing the formula: \(a*f(b(x-c))+d\)

Here are the steps I took to get from \(f(x) = \sqrt{x}\) to \(g(x)\).

  1. \(4\sqrt{x}\)
  2. \(4\sqrt{3x}\)
  3. \(4\sqrt{3(x-2)}\)
  4. \(4\sqrt{3(x-2)+3}\)

Numeric Algebraic Graphic Connection

I’ve included a graph of the functions described in this artifact. This graph backs up my claims. It is the visual/numerical representation of my algebraic formulas.

Appropriate Use of Technology

I used an online graphing calculator to generate the graph below.

Once I generated it:

  • I took a screenshot of the online graph
  • I cropped the screenshot
  • I added the image to my local code repository
  • I included the image in my source code
  • I uploaded the image to my code repository (https://github.com/doubledubba/precalc) and updated my code
  • I synchronized my readthedocs.org project with my repo

Artifact:

Transform \(f(x) = \sqrt{x}\) into \(g(x)\) with the following transformations:

  1. Vertical stretch by magnitude of four
  2. Horizontal shrink by magnitude of \({1 \over 3}\)
  3. Horizontal shift of two units
  4. Vertical shift of three units.

\(f(x) = \sqrt{x}\) (red)

\(g(x) = 4\sqrt{3(x-2)}+3\) (blue)

../_images/2_3.png

All graphical transformations by using each type of transformation

Source: I made it up.

Explanation:

This artifact demonstrates all graphical transformations by using each type of transformation.

It shows proficiency in:

  • Reflection
  • Translation
  • Stretches and shrinks

Artifact:

Transform \(f(x) = |x|\) into \(g(x) = -3|4(x+4)|-7\)

  1. Reflect over x-axis
  2. Vertical stretch by magnitude of 3
  3. Horizontal shrink by magnitude of \({1 \over 4}\)
  4. Horizontal shift by -4 units.
  5. Vertical shift by -7 units.