17.2 - Circles

Links and useful resources

Equation of a circle: $(x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}$
In any graph, why is it that subtracting something from the coordinate variable ( $x$ or $y$ ), shifts the graph in the positive direction for that coordinate?
- The adjusted term is like an infiltrator, sitting in exactly the same place as the original variable would have been, but...
- It alters the value just before the rest of the expression gets a chance to work with it. The expression ends up computing all of its values as if the axis was shifted by that amount, so the resulting graph is shifted.
- A similar thing happens when you multiply a coordinate variable - If for example you double the $x$ coordinate, the effect is as if the x-axis is stretched by a factor of 2, and so the graph itself is squished to half the size in the x dimension.
- This is always true for any graph of any equation, as long as every instance of the variable is modified in the same way.

(a) Find an equation whose graph is a circle with center $(4, - 5)$ and radius $3 \sqrt{2}$ .
Explain why the graph of the equation $(x - h)^{2} + (y - k)^{2} = r^{2}$ where $h$ , $k$ , and $r$ are constants with $r > 0$ , is a circle with center $(h, k)$ and radius $r$ .
What is the center and radius of the graph of $x^{2} + 2 x + y^{2} - 6 y = 6$