Circles Page 6

Page 6

Varying h, k and r in the Standard Equation of a Circle

Let's explore what happens to the graph of a circle as we vary h, k and r in the equation (x - h)² + (y - k)² = r² :

Use the action figure below and the "up" and "down" arrows to explore what happens to a circle when:
the value of r increases or decreases.
only the value of h increases or decreases.
only the value of k increases or decreases.
the values of h and k both increase or decrease.

Once you've explored these scenarios, use the questions below to check your understanding of varying h, k and r in the equation (x - h)² + (y - k)² = r².

(x - h)² + (y - k)² = r²

Did you notice that a circle exists for both positive and negative values of r even though we refer to r as the radius which must be a postive number?
The explanation for this is that the circle equation, x² + y² = r², considers r², and algebraically, both -r and r will satisfy this equation. However, geometrically r represents a distance and thus must be a positive number. We call r the length of the radius.