Graphing systems of inequalities

Systems of inequalities are similar to the system of regular equations.  It is composed of few equations and it has answers that satisfy the equations.  We first graph the indevidule graphs in the system.  secondly, we need to determine which side of the graph satisfy the inequality.  We can determine this by testing a point on the graph.  The final answer will be the area on the graph which all graphs overlap. 

Cramer's rule

Cramer's rule helps to make solving systems if equations easier.  Because the variables are eliminated and we can just deal with numbers.  First, we put coeficients in matrix form and we take the determinate of that.  And we find Dx or Dy by placing the answer colume to the variable. We take determinate of DY and Dx.  Thus, x=Dx/D and same for other variables.

Systems of equations

When we have a systems of equation, we can solve it algebraically and also solve with geometry we can graph or use substitution/elimination.  When we have exactly one answer, it means the system is consistent and independent. That means the lines intersect at exactly one point. When we have a infinete number of solutions, it means he lines overlap each other so every point on the lines is a solution. That is called consistent nod dependent.  When we have no solution, which is when lines are parallel toeach other. There is no intersection, so there is no solution.  Then the system is inconsistent.   

My polar equation pic

Polar coordinates.

This week we learned about pole cordintes.  Those are points on a circular graph.  It contains a r and a angle value. When graphing, if the angle is positive, we look counterclockwise, and if the angle is negative, we count clockwise. We could switch polar and rectangular coordinates back and force. From polar to rectangular, we use x=rCosø and y=rsinø to get x and y value 

Parabola

This week we talked about prabola. Prabola is a u shape graph either going up and down or left or right. Parabolas have derixtrice.  It also has a focus.  The directrix for vertical graphs is y=k-c and for horizontal graph is x=h-c.  The prabola is symmetric, and there is a axis of symmetry.  The equation for parabola is (x-h)^2=4c(y-k).  If the graph is side to side, the equation's x and y switches.  The focus is (h, k+c) and vertices is (h,k).   

rotating conic

When we graph a conic graph that is tilted, we could try to rewrite it back into the normal equation.  The equations have angles, to rewrite this equations, we first need to find the angle.  Cot2x=A-C/B.    We get the A,B,C values from the equation given.  Ax^2+Bxy+Cy^2+Dx+ey+F=0. He second step is to replace x and y with x'cosø-y'sinø and x!sinø+y'cosø.  The last step is to plug in the x and y into the equation. After plugging in, we just do algebra and simplify the new equation.   And now we have a new equation for the rotated conic graph.