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Transcript
FTCE Middle Grades Math 5-9 Solving Linear Systems of Equations by Graphing Graph both equations on one set of axes. The point of intersection is the solution of the system. Skills 6.13-6.16 and 6.19 Example: Solve the following linear system of equations using the elimination method. 4 x 2 y 4 1. x y 4 Multiply the second equation by 4 so that the coefficients of the x terms are opposites. 4 x 2 y 4 2. 4 x 4 y 8 Add the equations together. 4 x 2 y 4 Solving Linear Systems of Equations by Substitution 1. 2. 3. Solve one of the equations for one of the variables. Substitute what you found for the chosen variable back into the other equation and solve for the other variable. Solve for the first variable. 4 x 4 y 16 6 y 12 3. Example: Solve the following linear system of equations using the substitution method. 1. 2. y2 Solve for x by substituting the value found for y into one of the original equations. 4 x 2 y 4 4 x 2( 2) 4 4 x 2 y 4 4 x 4 4 x y 4 4 x 8 Solve for y in the second equation to get y x 4 . Substitute into first equation for y and solve for x. x 2 The solution is (-2, 2). Graphing Quadratic Functions y ax 2 bx c 4 x 2( x 4) 4 4 x 2 x 8 4 6 x 12 x 2 3. If a is positive, the parabola opens up. If a is negative, the parabola opens down. To graph a parabola: 1. Find the x-coordinate of the vertex using the Solve for y by substituting the value found for x. 4 x 2 y 4 4( 2) 2 y 4 8 2 y 4 2y 4 formula x 2. 3. y2 The solution is (-2, 2). Solving Linear Systems of Equations by Elimination 1. 2. 3. Multiply or divide either or both of the equations by a number so that the coefficients of one variable are opposites of each other. Add the equations to eliminate one of the variables. Substitute what you found for the chosen variable back into the other equation and solve for the eliminated variable. 4. b . 2a To find the y-coordinate of the vertex, substitute the x-coordinate of the vertex into the function. To find additional points on the parabola, pick x-values on both sides of the vertex and substitute into the function to find the corresponding y-values. Plot the points found in step 3 and connect with a smooth curve. Graphing Quadratic Inequalities 1. 2. 3. 4. Graph as if it is a quadratic equation. Use a dotted line for > or < symbols. Use a solid curve for ≥ or ≤ symbols. Select a point inside the parabola and substitute the x and y values of that point into the function. If the point is a solution of the inequality, shade inside the parabola. If the point is not a solution, shade outside of the parabola. FTCE Middle Grades Math 5-9 Factoring Skills 6.13-6.16 and 6.19 Factoring Trinomials with a Leading Coefficient other than 1: Factor out GCF: 1. 2. 3. 4. Find GCF of coefficients. Find smallest power of each variable. Multiply the terms from steps 1 and 2, this is the GCF. Divide each term in the polynomial by the GCF. 1. 2. 3. 4. Factor out the GCF if possible. Multiply leading coefficient and 3rd term. Find the factors that add to middle term. Split the middle term. Factor by grouping. Example: Factor the polynomial: 2 x 2 y 3 z 4 4 xyz 3 10 x 4 y 2 z 2 2 xyz 2 ( xy 2 z 2 2 z 5x 2 y ) Factor by Grouping Factoring by grouping is usually done when you have 4 terms and they do not look to have anything in common. 1. 2. 3. 4. Remove the GCF if possible. Group terms with common factors. Remove the GCF of each group. Use the distributive law to rewrite the expression. Example: Factor Example: Factor the trinomial: 2 x x 6 2 No GCF. Multiply 2 x 6 = 12. What two factors of 12 add up to 1? 4 and -3. Split the middle: 2 x 2 4 x 3x 6 (Notice the polynomial has not changed) Factor by grouping: 2 x 2 4 x 3x 6 2 x x 2 3( x 2) 2 x 3 x 2 6x2 9x 4x 6 Special Factoring Patterns: 6x 9x 4x 6 2 3x( 2 x 3) 2( 2 x 3) (3x 2)( 2 x 3) Factoring Trinomials with a Leading Coefficient of 1: 1. 2. Factor out the GCF if possible. Take factors of 3rd term that add up to the middle term and factor into 2 binomials. Example: Factor the trinomial: x2 x 6 ( x 3)( x 2) Difference of Squares: a b Difference of Cubes (SOAP): 2 2 (a b)(a b) a3 b3 (a b)(a 2 ab b2 ) Sum of Cubes (SOAP): a3 b3 (a b)( a 2 ab b2 ) SOAP: Same, Opposite, Always Positive FTCE Middle Grades Math 5-9 Skills 6.13-6.16 and 6.19