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Name__________________________________ Midterm Date: __________________________ Advanced Algebra II – Midterm Review Important Formulas and Tips Chapter 1 – Equations and Inequalities SOLVING LINEAR EQUATIONS - Distribute if needed - Combine like terms - Isolate your variable and solve SOLVING LINEAR INEQUALITIES - Distribute if needed - Combine like terms - Isolate your variable - Remember to flip the inequality sign if you multiply or divide by a negative - Your solutions are usually graphed on a number line with shading. SOLVING ABSOLUTE VALUE EQUATIONS - Isolate the absolute value - Transform the original equation into two linear equations (one negative and one positive) - Solve each equation (you should have 2 answers) SOLVING ABSOLUTE VALUE INEQUALITIES - Isolate the absolute value - Transform the inequality into an equivalent inequality - Less Than becomes an AND written as one 3 part inequality: -c < ax + b < c -c ≤ ax + b ≤ c - Greater Than becomes an OR written as 2 inequalities: ax + b > c OR ax + b < -c ax + b ≥ c OR ax + b ≤ -c Chapter 2 – Linear Equations and Functions RELATIONS - A relation is a function where every input value has exactly one output value - Domain are the X values, Range are the Y values - Vertical line test: A relation is a function if and only if no vertical line intersects the graph of the relation at more than 1 point LINES - To find the slope through 2 points: m y2 y1 x2 x1 - Horizontal lines - have slope of 0 and are the form y c . - Vertical lines - have an undefined slope and are in the form x c . - Perpendicular lines have slopes that are opposites and reciprocals - Parallel lines have the same slope - Slope-intercept form: y mx b - Point-slope form: y y1 m( x xz ) - use when given a point and the slope - To find the x-intercept of a line, plug in 0 for y and solve for x: (#, 0) - To find the y-intercept of a line, plug in 0 for x and solve for y: (0 , #) or (0, b) ABSOLUTE VALUE - Absolute value functions are of the form: y a | x h | k , and makes a “v”-shaped graph. - Vertex is (h, k), remember to switch the sign of your h - When graphing, plot your vertex, then use ‘a’ as your slope to plot the next point, use symmetry to get the other side of the graph. GRAPHING LINEAR INEQUALITY - Determine if the line is dashed or solid (depending on the inequality symbol) - Graph the line using either slope-intercept form or the x- and y-intercepts. - Test the point (0, 0) if you can to determine where to shade. If the test point is a solution, shade where the point lies, if it is not, shade where the point is not. Chapter 3 – Linear Systems SOLVING LINEAR SYSTEMS 1.) Substitution method: - Solve one of the equations for one of its variables - Substitute the expression from step 1 into the other equation and solve for the other variable - Substitute the value from step 2 into one of the original equations and solve. 2.) Elimination method: - Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables - Add the revised equations from step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable - Substitute the value in step 2 into either of the original equations and solve for the other variable GRAPHING SYSTEMS OF LINEAR INEQUALITIES - Graph each inequality in the system - Identify the region that is common to all the graphs, (where the shading overlaps, darken in this area) Chapter 4 – Quadratic Functions and Factoring GRAPHING QUADRATIC FUNCTIONS Standard form y ax 2 bx c b , then plug in x to find y 2a 2. Type the function into your graphing calculator to find a table of values. Look for symmetry! 1. To find the vertex: x Vertex form y a( x h)2 k 1. The vertex is (h, k), remember to flip the sign of h 3. Type the function into your graphing calculator to find a table of values. Look for symmetry! Intercept form y a( x p)( x q) 1. The x-intercepts are (p, 0) and (q, 0), remember to flip the signs of p and q pq 2. To find the vertex: x then plug in x to find y. (Average the x-intercepts!) 2 Additional Topics: Axis of symmetry: x-coordinate of the vertex. Maximum/minimum: corresponds to the y-value of the vertex. If the graph opens up, the vertex is a minimum, if the graph opens down the vertex is a maximum. Domain: (all real numbers) Range: Dependent on the y-value of the vertex and the orientation of the graph. 4.3 and 4.4 : Factoring Quadratics When the a value is 1, find the factors of c that add to b and use these factors in your binomials. Factoring Steps: 1.) Factor any common monomials (GCFs) 2.) If it is a Binomial: a.) Difference of Squares a 2 b2 (a b)(a b) Example : 4 x2 9 (2 x 3)(2 x 3) b.) If not a difference of 2 squares, then you are finished 3.) If it is a Trinomial: a.) Perfect Square Trinomial a 2 2ab b2 (a b)2 Example : x2 6 x 9 ( x 3)2 OR a 2 2ab b2 (a b)2 Example : 4 x 2 20 x 25 (2 x 5)2 b.) Guess and check using FOIL with the factors and signs. OR c.) Systematic method using factoring by grouping. **When solving, set each factor equal to zero and solve for x. ** 2.) Square Roots Vocab: Conjugates a + √𝑏 and a - √𝑏 Product Property: √𝑎𝑏 = √𝑎 √𝑏, when simplifying, you want the radicand (number under square root symbol), to have no factors which are a perfect square. 𝑎 Quotient Property: √𝑏 = √𝑎 √𝑏 , CANNOT HAVE A SQUARE ROOT IN THE DENOMINATOR, YOU MUST RATIONALIZE. (SIMPLIFY) Form in denominator √𝑏 Multiply numerator and denominator by √𝑏 a + √𝑏 a - √𝑏 a - √𝑏 a + √𝑏 When solving quadratics using square roots: 1). Isolate the squared term 2). Take the square root of both sides (DON’T FORGET TO PUT ± IN FRONT OF YOUR SQUARE ROOT) 3). Simplify the radical completely Complex Numbers (a + bi) √−𝟏 = i, i2 = -1 - When solving quadratics, do as you normally would, isolate the square term, take the square root of both sides (DON’T FORGET THE ±) - Write the square root in terms of i - Simplify Operations with Complex Numbers -When adding and subtracting, combine all real numbers, combine all complex numbers, write in standard form (a + bi) - When multiplying, simplify by distributing or foiling, write in standard form as combining like terms (don’t forget to simplify with i 2 1 ). - When dividing, multiply top and bottom by the complex conjugate (a + bi a – bi). FOIL or distribute in the numerator and multiply first and last in the denominator. (don’t forget to simplify with i 2 1 ). 3.) Quadratic Formula and Discriminant -QUADRATIC FORMULA: 𝑥= −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 , use when an equation cannot be factored. -DISCRIMINANT b2 – 4ac, the discriminant can determine the number and type of solutions for an equation. Value of discriminant b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0 Number and type of solutions 2 real solutions One real solution Two imaginary solutions Graph of Y = ax2 + bx + c Crosses x – axis 2 times Cross x-axis 1 time (at the vertex) Never crosses the xaxis. GRAPHING QUADRATIC INEQUALITIES - Determine whether the parabola is dashed or solid b , plot the vertex, use your calculator to find other 2a points to plot by using your table, connect the points with your dashed or solid parabola curve - Graph the quadratic by finding the vertex, x - Test the point (0,0) if possible. If it is a solution, shade where the test point IS, if the point is not a solution, shade where the test point is NOT GRAPHING SYSTEMS OF QUADRATIC INEQUALITIES - Graph each quadratic separately (see above) - Shade/darken in the region where they overlap SOLVING QUADRATIC INEQUALITIES ALGEBRAICALLY - Write and solve the equation by replacing the inequality with an equal sign. Then solve the equation. - Write your answer a compound inequality (either an and or an or statement). If the original inequality sign was less than then the compound inequality is an “AND” statement. If the original inequality sign was greater than then the compound inequality is an “OR” statement. *Note: Your conclusion statement should make sense based on the correct order of a number line. WRITING QUADRATIC MODELS Vertex form: y a( x h)2 k – used when you know the vertex and a point. - Plug in the given vertex for h & k - Plug in the point you are given for x and y - Solve for a - Write the equation including the values for a, h and k Intercept form: y a ( x p )( x q ) – used when you know the x-intercepts and a point. - Plug in the given x-intercepts for p and q - Plug in the point you are given for x and y - Solve for a - Write the equation including the values for p, q and a