Algebra 2 Midterm

4-5: Quadratic Equations

* The Quadratic Formula*

4-8: Complex Numbers

*Adding Subtracting Complex Numbers* 1.) Change all imaginary numbers to bi form. 2.) Add (or subtract) the real parts of the complex numbers. 3.) Add (or subtract) the imaginary parts of the complex numbers. 4.) Write the answer in the form a + bi.

4-6: Completing the Square (Cont.)

*Completing the Square* Step 1.) Divide all terms by a (the coefficient of x2). Step 2.) Move the number term (c/a) to the right side of the equation. Step 3.) Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation. We now have something that looks like (x + p)2 = q, which can be solved rather easily: Step 4.) Take the square root on both sides of the equation. Step 5.) Subtract the number that remains on the left side of the equation to find x. *link used*: https://www.mathsisfun.com/algebra/completing-square.html

2-7: Graphing Absolute Value Functions

*Describe the translations of absolute value functions* Parent Function: y=|x| Vertical Translation - Translation up K units; k>0// y=|x|+k - Translation down K units; k<0// y=|x|-k Horizontal Translation - Translation up H units; h>0// y=|x-h| - Translation down H units; h<0// y=|x+h| Vertical Stretch and Compression - Vertical stretch; a>1// y=a|x| - Vertical compression; 0<a<1// y=a|x| Reflection -In the x-axis// y=-|x| -In the y-axis// y=|-x|

2-1: Relations and Functions

*Determine if a relation is a function* (go to Algebra 2 Common Core textbook pg. 62)

4-4: Factoring Quadratic Equations

*Factor expressions* Factoring quadratics finds the roots or x-intercepts of a quadratic equation. Factoring quadratic equations in standard form, ax^2+bx+c, can often be accomplished by finding two numbers that add to give b, and multiply to give ac. In more complicated equations, use the quadratic formula or completing the square methods.

4-2: Standard Form of a Quadratic Function (Cont.)

*Find axis of symmetry, vertex, y-intercept, x-intercepts, domain, range* Ex: x^2+8x+12 x = −b/2a gives the x coordinate for the vertex x = −8/2 = −4 When x = −4y = [−4] 2 + 8 × − 4 + 12 y =16 − 32 + 12 = − 4 The vertex is (-4,-4) If we factor the equation y = (x + 2) (x + 6) So the x intercepts are when y = 0 ⇒ (x + 2) (x + 6) = 0 x = − 2 or x = − 6 Minimum (-4,-4) Axis of symmetry x = − 4 y intercept (0,12) x intercept (-2,0) and (-6,0) Domain (− ∞, ∞) Range #[-4, 00)

4-6: Completing the Square

*Find the c value that makes a perfect square* Ex: x^2 + 16x + c 1.) Find half of the coefficient of x.// 16/2 = 8 2.) Square the result of Step 1.// 8^2 = 64 3.) Replace c with the result of Step 2.// x^2 + 16x + 64 - then x^2 + 16x + 64 = (x+8)(x+8) = (x+8)^2

2-3: Linear Functions and Slope Intercept Form

*Find the slope with two points*

4-2: Standard Form of a Quadratic Function

*Find the vertex and all transformations to the parent function* The transformation of the parent function is shown in blue. It is a shift down (or vertical translation down) of 1 unit. A reflection on the x-axis is made on a function by multiplying the parent function by a negative. Multiplying by a negative "flips" the graph of the function over the x-axis. *link used*: https://www.anderson5.net/cms/lib02/SC01001931/Centricity/Domain/2147/PFTransformations%20with%20notes.ppt

2-4: More About Linear Equations

*Find the x and y intercept of an equation* To find the x-intercept of a given linear equation, plug in 0 for 'y' and solve for 'x'. To find the y-intercept, plug 0 in for 'x' and solve for 'y'

3-3: Systems of Inequalities

*Graph a system of inequality* To graph a linear inequality in two variables (say, and ), first get alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line. If the inequality is strict ( or ), graph a dashed line. *link used*: https://www.varsitytutors.com/hotmath/hotmath_help/topics/graphing-systems-of-linear-inequalities

2-7: Graphing Absolute Value Functions

*Graph and Solve absolute value functions* Ex: pg 110 1.) Identify the vertex // the vertex is at (-1,4) so h=-1 and k=4 2.) Identify a//the slope of the branch to the right of the vertex is -1/3, so a=-1/3 3.) Write the equation// substitute the values of a, h, and k into the general form y= a|x-h|+k. The equation that describes the graph is y= -1/3|x+1|+4

4-9: Quadratic Systems

*Graph to find the system of equations/ inequalities* *link used* https://www.youtube.com/watch?v=CgH4p2CyIHQ (Couldn't find step-by-step explanation)

4-8: Complex Numbers (Cont.)

*Multiplying Complex Numbers (FOIL)* 1.) Change all imaginary numbers to bi form. 2.) Multiply the complex numbers as you would multiply polynomials. 3.) Substitute -1 for each i2. 4.) Combine the real parts and the imaginary parts. 5.) Write the answer in the form a + bi.

1-6: Solving Absolute Value Inequalities

*Solve and graph absolute value equations. Check for extraneous solutions* 1.) Isolate the absolute value expression on the left side of the inequality. 2.) If the number on the other side of the inequality sign is negative, your equation either has no solution or all real numbers as solutions. 3.) Remove the absolute value bars by setting up a compound inequality. 4.) Solve the inequalities. *If your problem has a greater than sign (your problem now says that an absolute value is greater than a number), then set up an "or" compound inequality that looks like this: (quantity inside absolute value) < -(number on other side) OR (quantity inside absolute value) > (number on other side) The same setup is used for a ³ sign. If your absolute value is less than a number, then set up a three-part compound inequality that looks like this: -(number on other side) < (quantity inside absolute value) < (number on other side)*

4-4: Factoring Quadratic Equations (Cont.)

*Solve by factoring* 1.) Step 1: Write the equation in the correct form. To be in the correct form, you must remove all parentheses from each side of the equation by distributing, combine all like terms, and finally set the equation equal to zero with the terms written in descending order. 2.) Step 2: Use a factoring strategies to factor the problem. 3.) Step 3: Use the Zero Product Property and set each factor containing a variable equal to zero. 4.) Step 4: Solve each factor that was set equal to zero by getting the x on one side and the answer on the other side.

4-9: Quadratic Systems (Cont.)

*Solve by substitution* Ex:{y = x^2 - x + 6, y = x + 3} 1.) Substitute x + 3 in the quadratic equation 2.) Write in standard form 3.) Factor. Solve for x 4.) Substitute for x in y = x + 3 (Go to Algebra 2 Common Core pg. 259)

1-4: Solving Equations

*Solving A Multi-Step Equation*

1-4: Solving Equations (Cont.)

*Solving Literal Equations*

1-5: Solving inequalities and Graphing

*Solving and Graphing Multi-Step Equations* *linked used*:https://www.youtube.com/watch?v=FsrlQYXxbEQ (Couldn't find step-by-step explanation)

1-6: Solving Absolute Value Inequalities (Cont.)

*Solving and graphing absolute value inequalities*

1-6: Solving Absolute Value Inequalities (Cont.)

*Solving and graphing compound inequalities*

3-2: Solving Systems Algebraically

*Solving by Substitution and Elimination. Solving Word Problems* /Solving by Substitution/ 1.) solve one equation for one of the variables 2.) substitute the expression for y in the other equation. solve for x 3.) substitute the value for x into one of the original equations. solve for y /Solving by Elimination/ 1.) Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. ... 2.) Step 2: Subtract the second equation from the first. 3.) Step 3: Solve this new equation for y. 4.) Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x. /Solving a Word Problem/ Write a system of equations to model the situation. Add the equations to eliminate the y-term and then solve for x. Substitute the value for x into one of the original equations to find y. Check your answer by substituting x and y into the original system.

4-5: Quadratic Equations (Cont.)

*Use the Discriminant to determine the number of solutions* discriminant: b^2-4ac

2-3: Linear Functions and Slope Intercept Form (Cont.)

*Write the slope intercept form of the equation of a given line* To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope. This is the value of m in the equation. Next, find the coordinates of the y-intercept--this should be of the form (0, b). The y- coordinate is the value of b in the equation. *link used*: https://www.mathplanet.com/education/algebra-1/formulating-linear-equations/writing-linear-equations-using-the-slope-intercept-form

2-3: Linear Functions and Slope Intercept Form (Cont.)

*Write the slope intercept form of the equation of the line through the given points* Formula: y=mx+b Ex: m=1/5 y-intercept= (0,-3) - use slope-intercept form// y=mx+b - Substitue m=1/5 and b=-3// y=1/5x+(-3) - Simplify// y=1/5x-3 (go to Algebra 2 Common Core textbook pg. 77)

Chapter 1: Expressions, Equations, And Inequalities

1-4: Solving Equations - solve multi step equations -solve literal equations 1-5: Solving inequalities and Graphing - Solve and graph multi-step inequalities 1-6: Solving Absolute Value Inequalities - Solve and graph absolute value equations. Check for extraneous solutions - Solving and graphing absolute value inequalities - Solving and graphing compound inequalities

Chapter 2: Functions, Equations, and Graphs

2-1: Relations and Functions - Determine if a relation is a function 2-3: Linear Functions and Slope Intercept Form - Find the slope with two points - Write the slope intercept form of the equation of the line through the given points - Write the slope intercept form of the equation of a given line 2-4: More About Linear Equations - Find the x and y intercept of an equation 2-7: Graphing Absolute Value Functions - Graph and Solve absolute value functions - Describe the translations of absolute value functions 2-8: Two Variable Inequalities

Chapter 3: Linear Systems

3-2: Solving Systems Algebraically - Solving by Substitution and Elimination. Solving Word Problems - Break even type word problems 3-3: Systems of Inequalities - Graph a system of inequality

Chapter 4: Quadratic Functions and Equations

4-1: Quadratic Functions and Transformations 4-2: Standard Form of a Quadratic Function - Find the vertex and all transformations to the parent function - Find axis of symmetry, vertex, y-intercept, x-intercepts, domain, range 4-4: Factoring Quadratic Equations - Factor expressions - Solve by factoring 4-5: Quadratic Equations - The Quadratic Formula - Use the Discriminate to determine the number of solutions 4-6: Completing the Square - Find the c value that makes a perfect square - Completing the Square 4-8: Complex Numbers - Adding Subtracting Complex Numbers - Multiplying Complex Numbers (FOIL) 4-9: Quadratic Systems - Graph to find the system of equations/ inequalities -Solve by substitution

4-1: Quadratic Functions and Transformations

If a>0, the parabola opens upward. The y-coordinate of the vertex is the minimum value of the function. If a<0, the parabola opens downward. The y-coordinate of the vertex is the maximum value of the function.

2-8: Two Variable Inequalities

The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. The boundary line is dashed for > and < and solid for ≤ and ≥. *link used*: https://www.mathplanet.com/education/algebra-1/linear-inequalitites/linear-inequalities-in-two-variables