Transformations Of All Functions¶
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 \(g(x) = 3 * \sqrt{(x - 1)} + 2\)
- Horizontal shift of f(x) by one unit beacuse c shifts the graph horizontally by d units.
- Vertical shift of f(x) by two units because d shifts the graph vertically by d units.
- 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.
- \(-\sqrt{x}\)
- \(-4\sqrt{x}\)
- \(-4\sqrt{3x}\)
- \(-4\sqrt{3(x-2)}\)
- \(-4\sqrt{3(x-2)+5}\)
Artifact:
Transform \(f(x) = \sqrt{x}\) into \(g(x)\)
- Reflect over the x-axis
- Vertical stretch by a magnitude of four
- Horizontal shrink by a magnitude of \({1 \over 3}\)
- Horizontal shift by two units.
- Vertical shift by five units.
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 \(g(x)\).
- \(4\sqrt{x}\)
- \(4\sqrt{3x}\)
- \(4\sqrt{3(x-2)}\)
- \(4\sqrt{3(x-2)+3}\)
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:
- Vertical stretch by magnitude of four
- Horizontal shrink by magnitude of \({1 \over 3}\)
- Horizontal shift of two units
- Vertical shift of three units.
\(f(x) = \sqrt{x}\) (red)
\(g(x) = 4\sqrt{3(x-2)}+3\) (blue)
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 \(g(x) = -3|4(x+4)|-7\)
- Reflect over x-axis
- Vertical stretch by magnitude of 3
- Horizontal shrink by magnitude of \({1 \over 4}\)
- Horizontal shift by -4 units.
- Vertical shift by -7 units.