Basic Algebra Review

Math Analysis: Course Study Guide Basic Algebra Review **Before we start looking at harder concepts, it is very important to make sure that you have mastered the basic algebraic rules and concepts because these properties will be applied to the harder concepts. ➢ Arithmetic Operations ○ xy + xz = x(y+z) ■ As you can see both of the terms on the left side of this equation have ‘x’ therefore we can factor it out. ○ 𝑥 𝑦 𝑎 𝑥𝑏 + 𝑎𝑦 𝑏 𝑦𝑏 + = ■ When you are adding or subtracting fractions, both terms must have a common denominator. As you can see here, we multiplied both the numerator and denominator of both terms by the other term’s denominator, so they can both be added. 𝑎 𝑎𝑥 𝑏 𝑏 ○ x( ) = ■ When multiplying and dividing fractions you do not need a common denominator ○ 𝑎 𝑏 ÷ 𝑥 𝑦 = 𝑎𝑦 𝑏𝑥 ■ When you are dividing fractions remember the phrase “keep, change, flip”. You have to keep the 1st term the same, change the division sign to multiplication, and flip the spots of the numerator and denominator of the 2nd term. ○ 𝑎+𝑥 𝑏 𝑎 𝑥 𝑏 𝑏 = + ■ When you are adding or subtracting 2 terms in the numerator, you can separate them if they share a common denominator ➢ Absolute value ○ Absolute value is the distance a number is from zero. If there is a negative number in absolute value brackets it will become positive. ○ |𝑥𝑦|= |𝑥||𝑦| ○ |−𝑥| = |𝑥| 𝑥 |𝑥| ○ |𝑦|=|𝑦| ➢ Exponent properties ○ xaxb = xa+b ■ When multiplying 2 terms with exponents you can add their powers ○ (xa)b = xab ■ When you are raising a term with an exponent to another power you have to multiply the 2 exponents ○ (xy)a = xaya ■ In this situation, you are raising the quantity to a power; so you have to distribute the exponents to both terms. ○ x-a = 1 𝑥𝑎 ■ If a term is being raised to a negative exponent, you can get rid of the negative exponent by moving it to the denominator and making the exponent positive. ■ This can also be done if it were the opposite and you had a fraction with a negative exponent in the denominator. ○ X0 = 1 ■ Any term raised to a power of zero is 1 ➢ Properties of Radicals ○ 𝑛 √𝑥 = 𝑥 1 𝑛 ■ When you have a term with a radical, you can get rid of the radical by taking the term out of the radical and raising it to 1 over the power of the radical. ○ 𝑥 𝑦 √ √𝑎 = 𝑥𝑦√𝑎 ■ In this situation, you can multiply the powers of both radicals ○ 𝑥 𝑥 𝑥 √𝑎𝑏 = √𝑎 √𝑏 ■ When you are taking the square root of either a multiplication or division expression, you can distribute the radical to each term. ➢ Distance formula ○ The distance formula is used when you have 2 points and you want to find the distance between them. ○ If you take the Pythagorean theorem ( a2 + b2 = c2), and then solve for c you will get the distance formula: ➢ Complex Numbers ○ 𝑖 = √−1 ○ 𝑖 2 = −1 ○ √−𝑥 = 𝑖 √−𝑥 , a ≥ 0 ➢ Logarithms ○ y = logbx ≡ x = by ■ Ex; log5625 = 4 ≡ 625= 54 ○ Natural log: lnx = logex ■ e = 2.718281828… ○ Common log: logx = log10x ■ Anytime there is no base on log function, you should assume the base is 10 ○ logbb = 1 ■ When the base and the x value are equivalent that function will equal zero ○ logb1 = 0 ■ Whenever the x value of the log function is 1, regardless of the base, the function equals 0. Why? Because any number raised to the zero power = 1. ○ logbbx = x ○ The domain of all logarithmic functions is always x>0 ➢ Factoring Formulas ○ Difference of Squares ■ x2 - a2 = (x+a)(x-a) ○ Sum of Cubes ■ (x+a)(x2 - ax + a2) ○ Difference of Cubes ■ (x-a)(x2 + ax + a2) ○ Quadratic formula ■ ax2 + bx + c ■ x= −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 ○ If x2 = a then x = ±√a ■ Remember, the square root of a term can be positive and NEGATIVE ➢ Slope Formulas ○ Slope-intercept ■ y = mx + b ● m = slope ; b = y-intercept ○ Slope ■ y2 - y1/ x2 - x1 ○ Point-slope ■ y = y1 + m ( x - x1) ➢ Equations of different types of functions ○ Parabola/Quadratic ■ ax2 + bx +c ■ y = a ( x - h )2 + k ● Opens up if a > 0 ● Opens down if a < 0 ● The vertex is ( h , k ) ○ Circle ■ (x - h)2 + (y - k)2 = r2 ● ‘r’ represents the radius ● The center of the circle is (h,k) ○ Ellipse ■ (𝑥−ℎ)2 𝑎2 + (𝑦−𝑘)2 𝑏2 =1 ● Like circles, the center of an ellipse is also ( h , k ) ● ‘a’ represents the vertices ‘a’ units to the left/right ● ‘b’ represents vertices ‘b’ units up/down ○ Hyperbola ■ (𝑥−ℎ)2 𝑎2 - (𝑦−𝑘)2 𝑏2 =1 ● This equation is very similar to the equation of an ellipse, but the sign is in the middle is minus ● The center is still ( h , k ) ● This equation shows that the graph opens left and right ■ (𝑥−ℎ)2 𝑏2 - (𝑦−𝑘)2 𝑎2 =1 ■ This equation shows that the graph opens up and down Now we are ready to begin looking at some new content.