Translate the following statements in coordinate points:
A: f(-1) = 1
B: h(2) = 7
C: g(1) = -1
And if you are gonna troll me that's bad because I don't know this, I have only 20 points left, and you're making me get assignment questions wrong
Answers
Step-by-step explanation:
As you explore, keep in mind that every graph in the eBook is the graph of a precise mathematical function! The graphs do not result from complicated, computer graphics equations or programmed instructions. Each graph has a clearly defined domain and range, meaning, one number input and one number output. Even the 3D closed surface graphs have a fourth variable that allows the graph to pass the “vertical line test”!
Imaginary numbers can be defined as running along the depth axis of a 3D coordinate system.
If you’re interested in the rationale for this or just wondering, “Why imaginary numbers”? Or, “What is an imaginary number?” See: Imaginary Numbers and Human Experience”.
The eBook organizes the various functions with i by the number of dimensions in the function.
A 3D function has three variables. Each variable is graphed on one of the three space axes. The functions have one input variable x and two output variables y + iz and are of the form:
The input variable, x, is graphed on the horizontal axis. The real part of the output variable, y, is graphed on the vertical axis and the imaginary part of the output variable, iz, is graphed on the depth axis. In this way, the functions produce three dimensional graphs of conics (eBook sec. 2), polynomials (sec. 4), elliptic and hyperelliptic curves (sec. 4.5) and helixes. Helix functions, in side view, graph dozens of the well-known historical curves (section 3). A conic 3D function example is below.
A 4D function, of which there are several froms, has four variables. When the 4D function is of the form:
the function will graph open surfaces and closed surfaces like the examples in ‘Highlights 3’ and in the gallery images. These are in eBook sections 12 and 13.
A 4D function of the form:
combines the rotating natural exponential function with complex natural logarithms. See below the intro to the 6-part blog “Euler’s Formula Upgraded” and eBook section 6. A variation of these functions generates the “Cardioid Orbits” seen in the slider at the top of this page. Click on the slider to see the full animation.
Of note, in sections 6 and 11, a 4D function of the form:
generates the rotating imaginary exponential function with complex imaginary logarithms. The geometric difference between the natural and the imaginary exponential and complex log is here.
Other 4D function forms and their applications:
algebraic function morphing, sec. 9
helix morphing, sec. 10
5D functions, in eBook sec.14, of the parametric form:
will generate circular surfaces like the “triangular torus” and “square torus” seen in the slider at the top of this page.