FunctionsPointsA rational CurveThis meansThis means we are parameterizing curves…Formulax2 + y2 = r2Slide 8x2 + y2 = 25Slide 10x2 + y2 = 15Slide 12(x-2)2 +(y-5)2 = 6.25(x-3)2 +(y-3)2 = 36(x+2)2 +(y+4)2 = 121(x+7)2 +(y-1)2 = 100If you expand the equation does it change the shape?Can you work backwards?x2+4x+y2+8y = 101circle centered at P(h,k) tangent to LSlide 21Slide 22Slide 23So what is the process…Pythagorean TheoremFind the equation of the circleFunctionsCollege Algebra Chapter 7Points•P(x,y)–the point P with coordinates (x,y)A rational Curve•a plane algebraic curve f(x, y) = 0•is rational if it can be parameterized by rational functions.This means•two rational functions x = u(t), y = v(t) such that•at least one of u(t), v(t) is not a constant and f(u(t), v(t)) is identically zero.This means we are parameterizing curves…•note all curves are not straight•y = mx + b•y = ax2 + bx + c•(x-h) 2 + (y-k) 2 = r2CirclesFormulax2 + y2 = r2Formula for Circle centered at the originCenter point (0,0)Radius = r(0,0)rrrr•Center = (0,0)•Radius = 5•Plot the center•Count & Plot the radius in 4 directionsx2 + y2 = 25(0,0)(0,5)(0,-5)(5,0)(-5,0)•Center = (0,0)•Radius = sqrt(15)•Plot the center•Count & Plot the radius in 4 directionsx2 + y2 = 15(3.9,0)(-3.9,0) (0,0)(0,3.9)(0,-3.9)•Center = (2,5)•Radius = 2.5(x-2)2 +(y-5)2 = 6.25•Center = (3,3)•Radius = 6(x-3)2 +(y-3)2 = 36•Center = (-2,-4)•Radius = 11(x+2)2 +(y+4)2 = 121•Center = (-7,1)•Radius = 10(x+7)2 +(y-1)2 = 100(x+7)2 +(y-1)2 = 100•x2 +14x + 49 + y2 – 2y +1 = 100•x2 +14x + y2 – 2y = 100 – 49 – 1•x2 +14x + y2 – 2y = 50If you expand the equation does it change the shape?•x2+4x+y2+8y = 101•x2+4x+ +y2+8y+ = 101101•but remember if you add to one side of an equation you must add to the other.•x2+4x+ +y2+8y+ = 101 + +•how do you fill in the blanks to get an equation for a circle?•(x – h)2 + (y – k)2 = r2Can you work backwards?•x2+4x+ +y2+8y+ = 101101•x2+4x+ +y2+8y+ = 101 + +•can you make the connection from the left side toto the right?•(x – h)2 + (y – k)2 = r2•(x + 2)2 + (y + 4)2 = 121x2+4x+y2+8y = 10122224444221616CV # 14circle centered at P(h,k) tangent to LSo what is the process…•You have a line and a point…•Identify the slope and the perpendicular slope•Identify the perpendicular line through the given point•Find the point of intersection of the two lines•Find the distance between the two points.CV # 9Pythagorean Theorem•a2 + b2 = c2•a2 = c2 - b2•a2 + b2 - c2 = 0Find the equation of the