You must transform the equation in order to identify the center point and the radii in order to sketch the graph.
Transformation involves completing the square as for a circle with a couple of extra steps.
4x^{2} + 9y^{2} – 16x + 90y + 205 = 0 | Given this equation in standard form where |
4x^{2} – 16x + 9y^{2} + 90y = -205 | Rearrange the terms, group like terms and move the constant to the opposite side of the equation. |
4(x^{2} – 4x + ___) + 9(y^{2} + 10y + ___) = -205 | Group “x” terms and “y” terms and factor out the coefficients of the squared terms. |
4(x^{2} – 4x + 4) + 9(y^{2} + 10y + 25) = -205 + (4 x 4) + (9 x 25) |
Complete each of the “squares” by adding the appropriate quantities and add like quantities to the opposite side of the equation. |
4(x – 2)^{2} + 9(y + 5)^{2} = 36 | Write perfect squares in factored form and combine the constants |
four times x minus two squared over thirty-six 4(x − 2)^{2} 36 + nine times y plus five squared over thirty-six 9y + 5)^{2} 36 = thirty-six over thirty-six 36 36 | Divide each term by 36 so the left side equals 1. |
x minus two squared over nine (x − 2)^{2} 9 + y plus five squared over four (y + 5)^{2} 4 = 1 | Simplify all terms! |
Use your own graph paper or go to Free Graph Paper (this will open a new window/tab in your browser. Close the window/tab to return to this lesson) and sketch the graph.
h = __________ k = __________ center point = __________
x-radius = __________ y-radius = __________
Now try to transform, if needed, and graph the following: