Inflection Trace of Y=ax⁴+bx³+cx²+dx+e

The Java construction below is a graph of a quartic function with equation Y=ax⁴+bx³+cx²+dx+e. Increase the size of your window so you can see the entire applet. It may take a little while to load, so please be patient. Once the applet is ready, you can change each of the coefficients of the function by dragging the red points along the horizontal lines. As you modify one of the coefficients, notice the patterns in the locus of inflection points. Click once on an animate button to obtain a broad range of points in the locus, and click once again to stop the animation. Click on the red X in the lower corner to clear the trace. Try all of the animations to observe the relationship between the locus of inflection points and the varying coefficient in the equation. Answer the questions below when you are ready.

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Findings

Identify the function that is produced by the locus of the inflection points from a quartic function when you modify each of its coefficients. Input your responses at the right and press the TAB key after each response.

coefficient that varies	type of relation	equation of relation**	show me the solution
a	Choose one vertical linear relation oblique linear function quadratic function cubic function quartic function other		ok, I give up
b	Choose one vertical linear relation oblique linear function quadratic function cubic function quartic function other		ok, I give up
c	Choose one vertical linear relation oblique linear function quadratic function cubic function quartic function other		ok, I give up
d	Choose one vertical linear relation oblique linear function quadratic function cubic function quartic function other	?1=  ?2=  ?3=	ok, I give up
e	Choose one vertical linear relation oblique linear function quadratic function cubic function quartic function other	?4=  ?5=  ?6=	ok, I give up

**Enter the relation with the following criteria:

• use lower case letters;
• use variables x and/or y in an equation format (e.g., y=3x+2, or x=5);
• do not use any blank spaces in your input (e.g., use y=3x+2 instead of y = 3x + 2);
• use a caret sign, ^, to represent an exponent (e.g., use y=3x^2 to represent y=3x²);
• type an expression with descending exponents (e.g., use y=3x^2+5x-7 instead of y=5x+3x^2-7 or y=3x^2-7+5x, etc.);
• use the constants a, b, and/or c, along with other constants in your response (if necessary) (e.g., y=ax^2-cx+2);
• use parantheses only with each fraction (e.g., one-half should be entered as (1/2), and one divided by the quantity ab should be entered as (1/(ab));
• leave fractions in simplest form (e.g., use (2/(3a))x instead of (6/(9a))x);
• place coefficients in front of the variable (e.g., use (c/2)x^2 instead of (cx^2/2 or x^2(c/2));
• do not include a multiplication sign with any of the coefficients (e.g., use y=3x instead of y=3*x); and
• do not use decimals (e.g., use y=(5/2)x instead of 2.5x);