A2 Class Notes
Unit 3- Systems of Equations
11/2 & 11/5: Solving 3 x 3 Systems of Linear Equations
*The goal is to solve for all 3 variables.
*Take Eq. 1 and write it twice.
*Place Eq. 2 and Eq. 3 under equation 1!
*Using the Add/ Eliminate method, get down to 2 equations with 2 variables (Eq. 4 & 5).
*Use these new equations (Eq. 4 & 5) and solve for one variable using the Add/ Eliminate Method.
*Plug the variable into either Eq. 4 or Eq. 5. Solve for a second variable.
*Take the 2 variables you found and plug back into Eq. 1 to find the third variable.
*Check all 3 variables into either Eq. 2 or Eq. 3.
*The goal is to solve for all 3 variables.
*Take Eq. 1 and write it twice.
*Place Eq. 2 and Eq. 3 under equation 1!
*Using the Add/ Eliminate method, get down to 2 equations with 2 variables (Eq. 4 & 5).
*Use these new equations (Eq. 4 & 5) and solve for one variable using the Add/ Eliminate Method.
*Plug the variable into either Eq. 4 or Eq. 5. Solve for a second variable.
*Take the 2 variables you found and plug back into Eq. 1 to find the third variable.
*Check all 3 variables into either Eq. 2 or Eq. 3.
10/29- 31: Solving Systems of Equations
GRAPHICALLY
*Put both equations in standard form.
*Use your calculator to come up with a table of values for each that you can plot and that encompass the solutions (the intersection points).
*Graph both equations.
*Name the solution set (where they intersect).
ALGEBRAICALLY
*Put both equations in standard form.
*Set the equations equal to each other. Put this equation in standard form.
*Solve for x (Factor, Formula, CTS)- there should be 2 values of x.
*Find the corresponding y-value for each x-value.
*Check on your calculator.
GRAPHICALLY
*Put both equations in standard form.
*Use your calculator to come up with a table of values for each that you can plot and that encompass the solutions (the intersection points).
*Graph both equations.
*Name the solution set (where they intersect).
ALGEBRAICALLY
*Put both equations in standard form.
*Set the equations equal to each other. Put this equation in standard form.
*Solve for x (Factor, Formula, CTS)- there should be 2 values of x.
*Find the corresponding y-value for each x-value.
*Check on your calculator.
UNIT 2:
10/18: Solving Higher Order Equations Algebraically
*The degree of your equation tells you how many solutions there are.
*Remember, solutions can be real or imaginary, but when you solve algebraically, you find all solutions to the equation!
*In order to solve higher order equations algebraically, you must put the equation in standard form, factor and solve for all roots.
*The degree of your equation tells you how many solutions there are.
*Remember, solutions can be real or imaginary, but when you solve algebraically, you find all solutions to the equation!
*In order to solve higher order equations algebraically, you must put the equation in standard form, factor and solve for all roots.
10/17: Solving Higher Order Equations Graphically
*The degree of your equation tells you how many solutions there are.
*Remember, solutions can be real or imaginary, so the number of solutions (degree of the equation) does not always have to match graphically (only real solutions will cross the x-axis).
*A solution graphically is called a root (where the function crosses the x-axis).
*Use your calculator to determine where the function crosses the x-axis (y = 0).
*You can determine the sign of "a" by referencing lines and parabolas.
*EVEN roots TOUCH the x-axis (you write them twice), ODD roots CROSS the x-axis (you write them once).
*The degree of your equation tells you how many solutions there are.
*Remember, solutions can be real or imaginary, so the number of solutions (degree of the equation) does not always have to match graphically (only real solutions will cross the x-axis).
*A solution graphically is called a root (where the function crosses the x-axis).
*Use your calculator to determine where the function crosses the x-axis (y = 0).
*You can determine the sign of "a" by referencing lines and parabolas.
*EVEN roots TOUCH the x-axis (you write them twice), ODD roots CROSS the x-axis (you write them once).
10/15 & 10/16: Focus & Directrix:
*A parabola is defined as: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
*p = the distance from the vertex to the focus or the distance from the vertex to the directrix.
*y = 1/(4p)(x - h)^2 + k is the vertex form of a parabola.
*Always graph a picture to determine the vertex, a, and p.
*A parabola is defined as: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
*p = the distance from the vertex to the focus or the distance from the vertex to the directrix.
*y = 1/(4p)(x - h)^2 + k is the vertex form of a parabola.
*Always graph a picture to determine the vertex, a, and p.
10/11: Converting Forms of a Parabola
*Standard form is y = ax^2 + bx + c
*Vertex form is y = a(x - h)^2 + k, where the vertex is (h, k).
*To convert from vertex to standard, simply multiply and combine.
*To convert from standard to vertex, divide by a, move c, then CTS! Factor and simplify!
*Standard form is y = ax^2 + bx + c
*Vertex form is y = a(x - h)^2 + k, where the vertex is (h, k).
*To convert from vertex to standard, simply multiply and combine.
*To convert from standard to vertex, divide by a, move c, then CTS! Factor and simplify!
10/5: Circles
*Ax^2 + By^2 + Cx + Dy + E = 0 is the General form of a circle.
*(x - h)^2 + (y - k)^2 = r^2 is Center- Radius form of a circle.
*In order to convert from General Form to Center- Radius form, you must complete the square TWICE!
10/4: Completing the Square
*To CTS and create perfect square trinomial, simply take half of b and square it.
*To use CTS to solve a quadratic equation, divide everything by the a term and move the c term to the other side. CTS and make sure whatever you add to the left side of the equation, you add to the right side of the equation. Take the square root of both sides (don't forget the + -). Solve for x!
10/13: Quadratic Formula
*This formula can be used to solve any quadratic equation in standard form.
*This formula is on the reference sheet and does not need to be memorized.
*Be careful to simplify your answers in simplest radical form.
*Nature of the roots (discriminant = b^2 - 4ac)
*If the discriminant is positive then the solutions are real. If that positive number is a perfect square, then the solutions are real and rational. IF that positive number is not a perfect square then the solutions are real and irrational.
*If the discriminant is negative, then the solutions are imaginary.
*If the discriminant is 0, then there is only 1 real & rational solution to the quadratic equation.
9/27 & 9/28: Factoring
*There are 4 types of factoring we will study: greatest common factor (GCF), the difference of 2 perfect squares (DOTS), trinomials, and grouping.
*Always check for a GCF first!
*If the expression is a binomial, you must use either GCF or DOTS
*If there are 4 terms, you must use GCF or Grouping
*If the expression is a trinomial, try GCF and then use grouping to factor the trinomial.
*If you have a trinomial where a > 1, split the trinomial up into 4 terms and then use factor by grouping (what 2 terms multiply to ac, but combine to b?).
*Ax^2 + By^2 + Cx + Dy + E = 0 is the General form of a circle.
*(x - h)^2 + (y - k)^2 = r^2 is Center- Radius form of a circle.
*In order to convert from General Form to Center- Radius form, you must complete the square TWICE!
10/4: Completing the Square
*To CTS and create perfect square trinomial, simply take half of b and square it.
*To use CTS to solve a quadratic equation, divide everything by the a term and move the c term to the other side. CTS and make sure whatever you add to the left side of the equation, you add to the right side of the equation. Take the square root of both sides (don't forget the + -). Solve for x!
10/13: Quadratic Formula
*This formula can be used to solve any quadratic equation in standard form.
*This formula is on the reference sheet and does not need to be memorized.
*Be careful to simplify your answers in simplest radical form.
*Nature of the roots (discriminant = b^2 - 4ac)
*If the discriminant is positive then the solutions are real. If that positive number is a perfect square, then the solutions are real and rational. IF that positive number is not a perfect square then the solutions are real and irrational.
*If the discriminant is negative, then the solutions are imaginary.
*If the discriminant is 0, then there is only 1 real & rational solution to the quadratic equation.
9/27 & 9/28: Factoring
*There are 4 types of factoring we will study: greatest common factor (GCF), the difference of 2 perfect squares (DOTS), trinomials, and grouping.
*Always check for a GCF first!
*If the expression is a binomial, you must use either GCF or DOTS
*If there are 4 terms, you must use GCF or Grouping
*If the expression is a trinomial, try GCF and then use grouping to factor the trinomial.
*If you have a trinomial where a > 1, split the trinomial up into 4 terms and then use factor by grouping (what 2 terms multiply to ac, but combine to b?).
UNIT 1:
9/21: Solving Radical Equations
*Solve like a regular equation by performing inverse (opposite operations).
*Be sure to isolate the variable first!!!
*The inverse of a square root is to square the term.
*Always check for extraneous roots! (Solutions that seem like answers, but do not check into original equation)
*Solve like a regular equation by performing inverse (opposite operations).
*Be sure to isolate the variable first!!!
*The inverse of a square root is to square the term.
*Always check for extraneous roots! (Solutions that seem like answers, but do not check into original equation)
9/18 & 9/20: Operations with Imaginary & complex Numbers
*i = the square root of -1
*Powers of i can be found (up to a power of 6) on your calculator.
*Creating an i-clock will be helpful in simplifying powers of i.
*Operations with imaginary and complex numbers are done the same way you complete operations with polynomials or radicals.
*A complex number MUST be written in a + bi form.
*i = the square root of -1
*Powers of i can be found (up to a power of 6) on your calculator.
*Creating an i-clock will be helpful in simplifying powers of i.
*Operations with imaginary and complex numbers are done the same way you complete operations with polynomials or radicals.
*A complex number MUST be written in a + bi form.
9/13 & 9/14- Operations with Radicals
*To ADD or SUBTRACT radicals, it's just like polynomials- you MUST have the same last name (or like radicals). Simply combine the outsides (if possible) and keep the radical.
*To MULTIPLY radicals, multiply OUTSIDES with OUTSIDES and INSIDES with INSIDES. Then simplify your solution.
*To ADD or SUBTRACT radicals, it's just like polynomials- you MUST have the same last name (or like radicals). Simply combine the outsides (if possible) and keep the radical.
*To MULTIPLY radicals, multiply OUTSIDES with OUTSIDES and INSIDES with INSIDES. Then simplify your solution.
9/12- Simplifying Radicals
*Simplest radical form means there are no other perfect squares (perfect cubes, etc.)left under the radical symbol (radicand).
*If you have variables, you must break the variable's exponent into a multiple of the index. For example, if you are breaking down the cubic root of x to the 8th, break x to the 8th into x to the 6th and x squared.
*Perfect cubes are 1, 8, 27, 64, etc. Example: Something times itself times itself (3)(3)(3) = 27.
*Simplest radical form means there are no other perfect squares (perfect cubes, etc.)left under the radical symbol (radicand).
*If you have variables, you must break the variable's exponent into a multiple of the index. For example, if you are breaking down the cubic root of x to the 8th, break x to the 8th into x to the 6th and x squared.
*Perfect cubes are 1, 8, 27, 64, etc. Example: Something times itself times itself (3)(3)(3) = 27.
9/11- Operations with Polynomials
*To ADD polynomials, like up your like terms and combine the coefficients.
*To SUBTRACT polynomials, be sure to distribute the negative to the second group and then follow the rules for ADDING polynomials.
*To MULTIPLY polynomials, use the FOIL method (punnett square or double distribute). Remember, when you multiply the same base, add exponents!!!
*To ADD polynomials, like up your like terms and combine the coefficients.
*To SUBTRACT polynomials, be sure to distribute the negative to the second group and then follow the rules for ADDING polynomials.
*To MULTIPLY polynomials, use the FOIL method (punnett square or double distribute). Remember, when you multiply the same base, add exponents!!!