Assignment 1

by Jeff Hall

8. Graph the equation

where |x| is the absolute value of x. Variations?

The initial equation creates this graph:

The equation creates a heart-shaped figure that crosses

the x- and y- axes at +1 and -1.

If you change the sign in front of the first y, you get this equation and graph:

The heart is the same size and shape, but it is now inverted.

Now let's change the absolute value part so that the y is always positive.

Starting with the original equation, I made the first y in the equation

be the absolute value of y, and removed the absolute value sign

from the second x. Here is the new equation and graph:

 

The heart rotates clockwise 90 degrees.

What would happen if I changed the sign in front of the absolute value of y?

At this point, I predict that changing the sign will invert the heart again.

 

 

As expected, the heart is inverted.

Let's try something different. What happens when you multiply

the middle portion of the equation by the rational number n?

Here is the equation:

where the value of n cycles from -5 through 5

 

Here is the graph of the heart with the n value cycling from -5 through 5.

Note that when n equals zero, the heart shape becomes a perfect circle.

Also note that the x- and y- intercepts never change. Only the length and width

of the heart changes.

What if you wanted to change the x- and y- intercepts?

Consider what happened when n equaled zero. Since the n would cancel out

the xy portion, the equation would essentially become this:

Using our knowledge of circles, we know that changing the 1 would change the radius of the circle.

Would the same be true of our original heart equation? Let's try it:

where n cycles between 0 and 10.

Be still my beating heart!
Finally, let's investigate what happens when both x and y are absolute values.

Let's start with the original equation and make the change:

 

Now let's see it in motion with this equation:

where n cycles between -10 and 2.

It is interesting to note that this equation can be a diamond, square, or circle depending on the n.

 

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