Maintenance script: Math-Syntax korrigiert; Merksätze und Hinweise als normalen Text formatiert

2026-09-16T06:02:58Z

Math-Syntax korrigiert; Merksätze und Hinweise als normalen Text formatiert

← Nächstältere Version		Version vom 16. September 2026, 08:02 Uhr
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	<math>g(x)=a\,f(b(x-h))+k.</math>		<math>g(x)=a\,f(b(x-h))+k.</math>

	The parameters have predictable effects. The value <math>h</math> shifts the graph horizontally, while <math>k</math> shifts it vertically. The factor <math>a</math> controls vertical scaling and reflects the graph in the x-axis when <math>a~~<~~0</math>. The factor <math>b</math> controls horizontal scaling by a factor of <math>1/\|b\|</math> and reflects the graph in the y-axis when <math>b~~<~~0</math>.		The parameters have predictable effects. The value <math>h</math> shifts the graph horizontally, while <math>k</math> shifts it vertically. The factor <math>a</math> controls vertical scaling and reflects the graph in the x-axis when <math>a<0</math>. The factor <math>b</math> controls horizontal scaling by a factor of <math>1/\|b\|</math> and reflects the graph in the y-axis when <math>b<0</math>.

	A common error is to read horizontal transformations with the same sign intuition as vertical ones. In <math>f(x-4)</math>, inputs must be four units larger to produce the old outputs, so the graph moves four units to the right.		A common error is to read horizontal transformations with the same sign intuition as vertical ones. In <math>f(x-4)</math>, inputs must be four units larger to produce the old outputs, so the graph moves four units to the right.
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	== Derivatives as a Graphing Bridge ==		== Derivatives as a Graphing Bridge ==

	For learners approaching calculus, derivatives explain why graphing rules work. A derivative describes instantaneous rate of change. Where <math>f'(x)~~>~~0</math>, the graph is increasing; where <math>f'(x)~~<~~0</math>, it is decreasing. Points where <math>f'(x)=0</math> are candidates for local extrema, though further analysis is needed.		For learners approaching calculus, derivatives explain why graphing rules work. A derivative describes instantaneous rate of change. Where <math>f'(x)>0</math>, the graph is increasing; where <math>f'(x)<0</math>, it is decreasing. Points where <math>f'(x)=0</math> are candidates for local extrema, though further analysis is needed.

	The second derivative gives information about concavity. Where <math>f''(x)~~>~~0</math>, the graph is concave up; where <math>f''(x)~~<~~0</math>, it is concave down. An inflection point can occur where the concavity changes.		The second derivative gives information about concavity. Where <math>f''(x)>0</math>, the graph is concave up; where <math>f''(x)<0</math>, it is concave down. An inflection point can occur where the concavity changes.

	[[File:Cubic graph special points.svg\|500px\|frameless\|center]]		[[File:Cubic graph special points.svg\|500px\|frameless\|center]]
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	== Exponential Growth and Decay ==		== Exponential Growth and Decay ==

	An exponential function can be written <math>f(x)=Ab^x</math> with <math>b~~>~~0</math> and <math>b\neq 1</math>. If <math>b~~>~~1</math>, the basic function grows; if <math>0~~<~~b~~<~~1</math>, it decays. The natural exponential model <math>f(t)=Ae^{kt}</math> is especially useful because the parameter <math>k</math> represents a continuous relative growth rate.		An exponential function can be written <math>f(x)=Ab^x</math> with <math>b>0</math> and <math>b\neq 1</math>. If <math>b>1</math>, the basic function grows; if <math>0<b<1</math>, it decays. The natural exponential model <math>f(t)=Ae^{kt}</math> is especially useful because the parameter <math>k</math> represents a continuous relative growth rate.

	When <math>A~~>~~0</math>, <math>Ae^{kt}</math> stays positive. For growth with <math>k~~>~~0</math>, the doubling time is <math>\ln 2/k</math>. For decay with <math>k~~<~~0</math>, the half-life is <math>\ln 2/\|k\|</math>.		When <math>A>0</math>, <math>Ae^{kt}</math> stays positive. For growth with <math>k>0</math>, the doubling time is <math>\ln 2/k</math>. For decay with <math>k<0</math>, the half-life is <math>\ln 2/\|k\|</math>.

	[[File:Exponential Functions.svg\|500px\|frameless\|center]]		[[File:Exponential Functions.svg\|500px\|frameless\|center]]
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	<math>y=A\cos(B(x-C))+D.</math>		<math>y=A\cos(B(x-C))+D.</math>

	The amplitude is <math>\|A\|</math>. The period is <math>2\pi/\|B\|</math>. The horizontal shift is <math>C</math>, and the midline is <math>y=D</math>. If <math>A~~<~~0</math>, the graph is reflected across its midline.		The amplitude is <math>\|A\|</math>. The period is <math>2\pi/\|B\|</math>. The horizontal shift is <math>C</math>, and the midline is <math>y=D</math>. If <math>A<0</math>, the graph is reflected across its midline.

	[[File:Sine Cosine Graph.svg\|500px\|frameless\|center]]		[[File:Sine Cosine Graph.svg\|500px\|frameless\|center]]
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	f(x)=		f(x)=
	\begin{cases}		\begin{cases}
	x+2, & x~~<~~0,\\		x+2, & x<0,\\
	x^2, & 0\leq x~~<~~2,\\		x^2, & 0\leq x<2,\\
	4, & x\geq 2.		4, & x\geq 2.
	\end{cases}		\end{cases}
	</math>		</math>

	At a boundary, distinguish open and closed endpoints. If the rule uses <math>x~~<~~a</math>, the point at <math>x=a</math> is excluded from that piece. If it uses <math>x\leq a</math>, the endpoint is included.		At a boundary, distinguish open and closed endpoints. If the rule uses <math>x<a</math>, the point at <math>x=a</math> is excluded from that piece. If it uses <math>x\leq a</math>, the endpoint is included.

	[[File:Piecewise linear function.svg\|500px\|frameless\|center]]		[[File:Piecewise linear function.svg\|500px\|frameless\|center]]
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	The absolute-value function is itself piecewise:		The absolute-value function is itself piecewise:

	<math>\|x\|=-x</math> for <math>x~~<~~0</math> and <math>\|x\|=x</math> for <math>x\geq 0</math>.		<math>\|x\|=-x</math> for <math>x<0</math> and <math>\|x\|=x</math> for <math>x\geq 0</math>.

	This explains why its graph is made from two line segments with different slopes.		This explains why its graph is made from two line segments with different slopes.

Glanz: aiMOOC über GPT aiMOOC Action erstellt

2026-08-28T06:55:07Z

aiMOOC über GPT aiMOOC Action erstellt

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English:Advanced Functions and Graphs - Versionsgeschichte

Maintenance script: Math-Syntax korrigiert; Merksätze und Hinweise als normalen Text formatiert

Glanz: aiMOOC über GPT aiMOOC Action erstellt