Graph of y² = f(x)


Exercises for students
Solve the following problems on paper and check your graphs using the above applet:

Given that f(x) = x^3 - 2x , sketch on separate diagrams the graphs of y = f(x), y = | f(x) | , y = sqrt(f(x)), y² = f(x)
Given that f(x) = 1/(x^2-1) , sketch on separate diagrams the graphs of y = f(x), y = | f(x) | , y = sqrt(f(x)), y² = f(x)
Given that f(x) = x*(1-x^2) , sketch on separate diagrams the graphs of y = f(x), y = | f(x) | , y = sqrt(f(x)), y² = f(x)
Given that f(x) = x/(1+x) , sketch on separate diagrams the graphs of y = f(x), y = | f(x) | , y = sqrt(f(x)), y² = f(x)


Summary
To sketch y = | f(x) |, reflect the part of the graph of y = f(x) below the x-axis about the x-axis.

To sketch y = f( |x| ), ignore the part of the graph of y = f(x) on the left of the y-axis, and reflect the part of the graph of y = f(x) on the right of the y-axis about the y-axis.

To sketch y = sqrt f(x), note:

• sqrt f(x) is defined only if f(x) >= 0
• sqrt(0) = 0 and sqrt(1) = 1
• sqrt f(x) < f(x) if f(x) > 1
• sqrt f(x) > f(x) if 0 < f(x) < 1

To sketch y² = f(x), draw y = sqrt f(x) and y = -sqrt f(x).

To sketch y = 1 / f(x), note:

• 1 / f(x) is undefined when f(x) = 0
• 1 / f(x) = 0 when f(x) has a vertical asymptote
• 1/1 = 1 and 1 / (-1) = -1
• As f(x) increases, 1 / f(x) decreases and vice versa
• If y = f(x) has a maximum point, then y = 1 / f(x) has a minimum point and vice versa

You can define the function using the following operators:

+	-	*	/	^
abs( )	sqrt( )	ln( )	exp( )	pi
	sin( )	cos( )	tan( )
	asin( )	acos( )	atan( )
	sinh( )	cosh( )	tanh( )

Right-click here to download this page, the Java Class File and unzip the Javathings Math Package.
This applet uses the com.javathings.math package developed by:
Patrik Lundin
patrik@javathings.com
http://www.javathings.com