There are opposite effects on the demand and supply with a change in price.
The demand curve is a downward sloping one and the supply curve is an upward sloping one. For an increase in price of a product the quantity that buyers are willing to buy of that product goes down. On the other hand the quantity that producers are willing to supply of the product goes up. The reverse happens when the price goes up, the demand from buyers increases and the supply from producers decreases.
The demand versus price graph and the supply versus price graph intersect at a point called the equilibrium point. When the price is lower than that at the equilibrium point, the supply would tend to decrease and demand would tend to increase and vice versa if the price were more than that at the equilibrium point.
If you are using economic terms precisely, a change in the price of a good or service does not affect the demand for or supply of that product. It will affect the quantity demanded or supplied but not the actual demand or supply. It will, in other words, cause a movement along a curve, but will not move the curve.
An actual change in supply or demand must be caused by some other factor. For example, people may come to demand more milk if a study comes out showing milk prevents cancer. There are ways in which the price of a good or service can impact demand or supply of another product. For example, an increase in the price of McDonald's burgers might cause people to demand more of what are called "competing goods." That is, they might demand more Burger King burgers or more pizza because the price of McDonald's has risen.
So, to be precise, the change in the price of a good cannot change the supply or demand for that good.