SAT Math Study Guide

SAT Math Study Guide SAT Test Information 2 Use of Calculator 3 Linear Equations 9 Solving Equations 1​1 Solving Inequalities 1​1 Systems of Equations 1​2 Nonlinear Equations 1​4 Solving Quadratic Equations 1​5 Polynomials 1​7 Rational Functions 1​8 Radicals and Exponents 20 Sequences 21 Scientific Notation 2​2 Scatterplots 2​2 Linear vs Exponential Growth 2​3 Probability 2​3 Center and Distribution 2​4 Writing Formulas in Terms of Variable 2​5 Surface Area and Volume Word Problems 2​6 Angles 2​6 Trigonometry 2​7 Complex Numbers 2​8 Sources 2​9 1 SAT Test Information Given Formulas Math Test- No Calculator ○ 25 Minutes for 20 Questions Math Test- Calculator ○ 55 Minutes for 38 Questions 2 Use of Calculator Programming Programming a graphing calculator can be an ​easy way to prepare for the test.​ It allows you to more quickly solve easy problems. Here are a ​few programs​ you can create for your calculator: *If you need any other programs you can download more off of the TI site. 1. Quadratic Formula Step 1:​ Press the [PRGM] button and move over to the right twice to Create New. Step 2:​ Name the program Quad. Step 3:​ Add ClrDraw to the program by going to [2ND] [PRGM]. Step 4:​ Add Input by going to [PRGM] moving to the Right by one and pressing Input. Step 5:​ Type in “A=”. The equal sign can be found by going to [2ND] [MATH] and pressing =. Step 6:​ After typing “A=” add a comma and an A. Step 7:​ Repeat Steps 4-6 twice more replacing A with B and C as shown. Step 8:​ Write the Quadratic formula as follows: ((-B+√(B²-4AC))/(2A)) 3 Step 9:​ Store that equation as P by pressing [STO➤]. Then write P. Step 10:​ Repeat the process in Steps 9-10 instead writing ((-B-√(B²-4AC))/(2A)) and storing it as N. Step 11:​ Add Disp by going to [PRGM] moving to the Right by one and pressing Disp. Step 12:​ Write “X=”, P. Step 13:​ Add Disp and write “X=”, N. Example: x2 + x − 6 Plug in numbers for A, B, and C. A=1 B=1 C=-6 After pressing Enter, you should get the following answer: X=2 And X=-3 2. Distance Formula 4 Step 1:​ Press the [PRGM] button and move over to the right twice to Create New. Step 2:​ Name the Program Distance. Step 3:​ Add 4 Inputs for X1, Y1, X2, and Y2 as shown in the picture. Step 4:​ Add the equation as follows: √((A-X)²+(B-Y)²) Step 5:​ Store the equation as D. Step 6:​ Add the Disp and write “DIST=”, D. Example: Find the distance between the points (3,4) and (-2, 5). Plug in the numbers for X1, Y1, X2, and Y2, X1= 3 Y1= 4 X2= -2 Y2= 5 After pressing Enter, you should get the following answer: DIST= 5.099019514 3. Slope Formula 5 Step 1:​ Press the [PRGM] button and move over to the right twice to Create New. Step 2:​ Name the Program Slope. Step 3:​ Add 4 Inputs for X1, Y1, X2, and Y2 as shown in the picture. Step 4:​ Add the equation as follows: (A-X)/(B-Y) Step 5:​ Store the equation as S. Step 6:​ Add the Disp and write “SLOPE=”, S. Example: Find the slope of a line using the points (3,4) and (-2, 5). Plug in the numbers for X1, Y1, X2, and Y2. X1= 3 Y1= 4 X2= -2 Y2= 5 After pressing Enter, you should get the following answer: SLOPE= -.2 6 Basic Use of Calculator 1. How to Graph an Equation Step 1:​ Press [Y=]. Step 2:​ Enter Equation. Step 3: Press [GRAPH] to see the graph or [2ND][GRAPH] to see the table. 2. Analyzing a Graph a. Calculating a value Step 1:​ Press [2ND][TRACE][1: value]. Step 2:​ Enter a value of X and press [ENTER]. b. Finding zeros Step 1:​ Press [2ND][TRACE][2: zero]. Step 2:​ Move left of the zero and press [ENTER]. Step 3:​ Move right of the zero and press [ENTER]. c. Finding minimum/maximum Step 1:​ Press [2ND][TRACE][3: minimum] or Press [2ND][TRACE][4: maximum]. Step 2:​ Move left of the minimum/maximum and press [ENTER]. Step 3:​ Move right of the minimum/maximum and 7 press [ENTER]. 3. Linear Regression/Line of Best Fit Step 1:​ Press [STAT][1: Edit…]. Step 2:​ Enter X and Y values. Step 3:​ Press [STAT][CALC][4: LinReg(ax+b)]. Step 4:​ Add Lists to Xlist and Ylist and press [Calculate]. 8 Linear Equations ● Slope-intercept form ○ y = mx + b ○ m is the slope and b is the y-intercept ● Point slope form ○ y − y 1 = m (x − x 1 ) ○ m is the slope and ( x 1 , y 1 ) is any point on the line ● Standard form ○ ax + by = c ○ *b in the Standard form is different from b in Slope-intercept form ● Vertical Equation ○ x=a ○ a is the x-coordinate throughout the whole line ● Horizontal Equation ○ y=b ○ b is the y-coordinate throughout the whole line ● Examples: ○ y = 3x + 5 ■ Slope: 3, y-intercept: 5 ○ y − 5 = 4 (x − 2) ■ Slope: 4, Point on Line: (2,5) ○ 2x + 5y = 12 9 ■ Slope: -⅖, y-intercept: 12/5 ○ x=2 ■ X-Coordinate: 2 ○ y=6 ■ Y-Coordinate: 6 10 Solving Equations Solving equations for a variable ○ Step 1:​ Move all like terms to the same side of the equation ○ Step 2:​ Combine like terms ○ Step 3:​ Solve for variable ● Example: ○ Equation: 4x + 1 = x − 5 ○ Step 1:​ 4x − x = − 5 − 1 ○ Step 2:​ 3x = -6 ○ Step 3:​ x = -2 Solving Inequalities Solving inequalities for a variable ● Similar to solving equations ○ Step 1:​ Move all like terms to the same side of the equation ○ Step 2:​ Combine like terms ○ Step 3:​ Solve for variable ● Don’t forget to change inequality sign when multiplying/dividing by a negative ● Example: ○ Equation: − 4x − 2 ≥ − 3x + 5 ○ Step 1:​ − 4x + 3x ≥ 5 + 2 ○ Step 2:​ − x ≥ 7 ○ Step 3:​ x ≤ − 7 ■ *Remember to flip the sign 11 Systems of Equations Setting up Systems of Equations from a word problem: ○ Example: Julie has $11.50 and wants to buy a total of 15 apples and oranges. Each apple cost $1.20 and each orange cost $0.70. How many apples and oranges does she have? ■ Step 1: Create/Assign variables ● a = apples ● o = oranges ■ Step 2: Set up 2 Equations ● a + o = 15 ○ Total apples and oranges ● 1.20a + 0.70o = 11.50 ○ Total cost of fruit ■ Step 3: Choose one of 3 Methods to Solve ● Substitution: ○ Plug in one equation into the other to solve for one variable. a + o = 15 and 1.2a + 0.7o = 11.5 a = 15 − o and 1.2a + 0.7o = 11.5 1.2(15 − o) + 0.7o = 11.5 18 − 1.2o + 0.7o = 11.5 18 − 0.5o = 11.5 − 0.5o =− 6.5 o = 13 12 a + 13 = 15 a=2 o = 13 and a = 2 ● Elimination: ○ Manipulate equations so one variable can be canceled out. a + o = 15 1.2a + 0.7o = 11.5 − 1.2 * (a + o) = (15) *− 1.2 1.2a + 0.7o = 11.5 − 1.2a − 1.2o = − 18 1.2a + 0.7o = 11.5 − 0.5o = − 6.5 o = 13 a + 13 = 15 a=2 o = 13 and a = 2 ● Graphing: ○ Graph both equations and see where they intersect. 13 ● All three methods come out to a = 2 and . Nonlinear Equations ● Quadratic Equation ○ ax2 + bx + c = 0 ○ y = a(x − h)2 + k ■ Vertex: (h,k) ■ If a > 0, the parabola opens up ■ If a < 0, the parabola opens down ● Exponential Equation ○ y = ex ● Logarithmic Equation ○ y = ln(x) ● Equation of Circle ○ (x − h)2 + (y − k )2 = r2 ○ r: radius ○ (h,k): center 14 ● Equation of Ellipse ○ 2 (x−h) a2 + 2 (y−k) b2 =1 ○ (h,k): center ○ a: horizontal distance from center to vertex ○ b: vertical distance from center to vertex ● Equation of Hyperbola ○ 2 (x−h) a2 − 2 (y−k) b2 =1 ■ Hyperbolas open to the right/left ○ 2 (y−k) a2 − 2 (x−h) b2 =1 ■ Hyperbolas open up/down ○ (h,k): center ○ a: the distance from the center to the vertex Solving Quadratic Equations ● Example: Solve for x2 + 4x − 12 = 0. Completing the Square ○ Step 1: Move the constant to the other side ■ x2 + 4x = 12 ○ Step 2: Complete the square by using ( b2 )2 ■ x2 + 4x + 4 = 16 ○ Step 3: Factor the expression ■ (x + 2)2 = 16 ○ Step 4: Square root both sides ■ √(x + 2) 2 = √16 = x + 2 =± 4 15 ○ Step 5: Solve for x ■ x =− 2 ± 4 =− 6 or 2 Factoring ○ Step 1: Multiply the constants of the first and last term ■ 1 *− 12 =− 12 ○ Step 2: Find 2 terms that multiply to get the previous step and that add to get the middle term ■ 6 *− 2 =− 12 and 6 + (− 2) = 4 ○ Step 3: Factor ■ (x + 6)(x − 2) ○ Step 4: Solve for each factor ■ x =− 6, 2 Quadratic Formula ○ ax2 + bx + c = 0 x= −b±√b2 −4ac 2a ○ x= −4±√42 −4(1)(−12) 2(1) ○ x= −4±√16+48 2 ○ ○ x =− 2 ± 4 = -6, 2 16 Polynomials ● Binomials​ have two terms. ● Trinomials​ have three terms. ● Polynomials​ have four or more terms. Adding/Subtracting Polynomials ○ Add or subtract like terms. ○ Example: ■ (3x2 + 2x − 6) − (2x2 − 6x + 3) ■ x2 + 8x − 9 Multiplying Polynomials ○ FOIL-​ First, Outside, Inside, Last ○ Example: ○ (3x+2)(4x-1) ○ 12x2 − 3x + 6x − 2 = 12x2 + 3x − 2 Dividing Polynomials ○ Example: ○ 2x2 −6x+4 x−2 ○ Polynomial Division 17 ○ Synthetic Division Rational Functions Adding/Subtracting Rational Functions ○ Add or subtract like terms. ○ They have to have the same denominator. ○ Example: ■ 3x2 −2x+5 3x − 2x2 +5x−6 3x = x2 −7x+11 3x ■ 4x2 +2x−7 x + 3x2 −6x+4 2x = 2(4x2 +2x−7) 2x + 3x2 −6x+4 2x = 11x2 −2x−10 2x Multiplying Rational Functions ○ Multiply denominator and numerator. ○ Example: ■ 2x+3 4x2 * x−4 2x3 = (2x+3)(x−4) 8x5 = 2x2 −5x−12 8x5 Dividing Rational Functions 18 ○ Flip the second rational and change the division sign to a multiplication sign. ○ Example: ■ 4x−5 2x2 ÷ 6x+1 4x3 = 4x−5 2x2 * 4x3 6x+1 = 2x(4x−5) 6x+1 = 8x2 −10x 6x+1 Finding Horizontal and Vertical Asymptotes ○ Vertical Asymptotes ■ Set the denominator equal to 0. ○ Example: ● x2 −3x+1 x+1 ● x+1=0 ● x =− 1 ○ Horizontal asymptotes ■ If the ​highest degree of the numerator equals the degree of the denominator​, divide the highest degrees to get the horizontal asymptote. ■ If the ​highest degree of the numerator is less than the degree of the denominator​, the horizontal asymptote is 0. ■ If the ​highest degree of the numerator is greater than the degree of the denominator​, there are no horizontal asymptotes. 19 Radicals and Exponents Exponent and Radical Rules Product Rule Quotient Rule xm * xn = xm+n xm xn = xm−n Power of Product Rule (x * y )n = xn * y n Power Rule (xm )n = xm*n Zero Exponent Rule (xm )0 = 1 Negative Exponents x−m = Adding/Subtracting Exponents 1 xm xm ± xn = xm ± xn xm + xm = 2xm m n Fractional Exponents/Radicals x n = √xm Multiplying Radicals √x * √y = √xy Dividing Radicals √x √y = √ x y 20 Sequences Arithmetic Sequences ○ Arithmetic sequences are when a ​common number is added or subtracted from each number. ○ Example: 1​ , 3, 5, 7 ■ +2 ○ Formula:​ a n = a 1 + (n − 1)d ■ a n = nth term in sequence ■ a 1 = first term in sequence ■ n = number in sequence ■ d = common difference ○ Example Problem: The first five terms of an arithmetic sequence are given: -7, -2, 3, 8, 13,… What is the eighth term in the sequence? ■ Solution: a 8 =− 7 + (8 − 1) * 5 a 8 =− 7 + (7) * 5 a 8 =− 7 + 35 a 8 = 28 Geometric Sequences ○ Geometric sequences are ​always multiplied or divided by the same number throughout the sequence. ○ Example:​ 1, 4, 16, 64 ■ *4 ○ Formula:​ a n = a 1 rn−1 ■ a n = nth term in sequence ■ a 1 = first term in sequence ■ r = common ratio 21 ■ n = number in sequence ○ Example Problem: The first term is 2 in the geometric sequence 2, -8, 32, -128…. What is the eleventh term of the geometric sequence? ■ Solution: a 11 = 2(− 4)(11−1) a 11 = 2(1048576) a 11 = 2097152 How to Tell the Difference ○ Arithmetic Sequences will always go up by the same amount while geometric sequences will go up by multiples. Scientific Notation ● A way to write large or small numbers using powers of 10s. ● Large Numbers ○ Form: x * 10n ○ Example: 284, 000, 000 = 2.84 * 108 ● Small Numbers ○ Form: x * 10−n ○ Example: 0.0000000284 = 2.84 * 10−8 Scatterplots Correlation ○ Strong Correlation ■ The line of best fit is close to ​most points. ○ Weak Correlation 22 ■ The line of best fit is close to ​some points. ○ No Correlation ■ The line of best fit is close to a ​few points. ○ Positive Correlation ■ The slope of the line of best fit is ​positive. ○ Negative Correlation ■ The slope of the line of best fit is ​negative. Linear vs Exponential Growth Linear ○ If the y changes by adding the same values, it represents ​linear growth. ■ Changes at a constant rate Exponential ○ If the y changes by multiplying the same values, it represents​ exponential growth. ■ Changes by a% ■ Halves/Doubles Probability Probability ○ AND Probability ■ Probability of multiple things happening. ■ P (A ⋂ B ) = P (A) * P (B) ■ Probability of A and B ○ OR Probability ■ Probability of one thing OR another thing happening. ■ P (A ⋃ B ) = P (A) + P (B) ■ Probability of A or B ○ Conditional Probability ■ Probability of something happening GIVEN something else happens. 23 ■ P (A|B) = P (A⋂B) P (B) ■ Probability of A given B. Independent vs Dependent ○ Independent Probability ■ Probability is NOT affected by another event happening. ○ Dependent Probability ■ Probability IS affected by another event happening. Permutations ○ Order matters ○ n P r= n! (n−r)! ■ n = total number of items ■ r = selected number of items Combinations ○ Order does not matter ○ nC r = n! r!(n−1)! ■ n=total number of items ■ r=selected number of items Center and Distribution Median ○ Number in the middle of a set of numbers ○ Example: ■ 1, 3, 6, 3, 7, 4, 2, 8, 1, 5 ■ Step 1:​ Reorder Numbers. ● 1, 1, 2, 3, 3, 4, 5, 6, 7, 8 ■ Step 2:​ Find the middle number (If there are 2 in the middle find the average of them). ● Median= 3+4 = 3.5 2 Mean ○ The average of all of the numbers 24 ○ Add all of the numbers and divide by the amount of numbers there are. ○ Example: ■ 1, 3, 6, 3, 7, 4, 2, 8, 1, 5 ■ Step 1:​ Add all numbers together. ● 1 + 3 + 6 + 3 + 7 + 4 + 2 + 8 + 1 + 5 = 40 ■ Step 2:​ Find out how many numbers there are in the set. ● There are 10 numbers. ■ Step 3:​ Divide the sum by the amount of numbers. 40 ● Mean= 10 =4 Mode ○ The number that shows up the most in the set ○ Example: ■ 1, 3, 6, 3, 7, 4, 2, 8, 1, 5 ■ Mode= 3 and 1 Standard Deviation ○ Standard deviation shows the spread of the data ○ The more spread out the data is, the ​greater​ the standard deviation is. ○ The less spread out the data is, the smaller the standard deviation is. Range ○ The difference between the largest and smallest number ○ Example: ■ 1, 3, 6, 3, 7, 4, 2, 8, 1, 5 ■ Step 1:​ Find the largest and smallest number. ● Largest= 8 and Smallest=1 ■ Step 2:​ Subtract the smallest number from the largest number. ● Range= 8-1=7 Writing Formulas in Terms of Variable Solving to get a variable alone on one side ● Example: ○ V = πr2 h The formula gives the volume V with a radius r and height h . Write r in terms of V and h . ○ Step 1: Get r alone on one side. 25 V ■ πh = r2 ○ Step 2:​ Solve for r . ■ r= √ V πh Surface Area and Volume Word Problems ● Most formulas for solids are given to you in the formula area of the math section. ● Always draw a Picture. ● Rectangular Prism ○ Volume= lwh ○ Surface Area= 2lw + 2lh + 2wh ○ Diagonal= √(l 2 + w2 + h2 ) Angles ● Parallel Lines ○ Lines that point the same direction and will never meet. ● Perpendicular Lines ○ Lines that meet at a 90° angle. ● Supplementary Angles ○ Angles that add up to 180°. ● Complementary Angles ○ Angles that add up to 90°. ● Opposite Angles ○ Angles opposite of each other when two lines intersect are equal. ● Remember that the interior angles of a triangle equal 180°. ● Example: What is the value of y 26 ○ Step 1:​ Find the value of x ■ X is the third value in the triangle, so you can use the equation 180 = 102 + 36 + x ■ 180 = 138 + x ■ x = 42 ○ Step 2:​ Find the value of y using x ■ The angle y is opposite the angle x, so the value of y will be the same as x ■ y=42 Trigonometry Right Triangle ○ Remember Soh Cah Toa to remember how to get sine, cosine, and tangent from a triangle. ■ Soh: S ine = opposite hypotenuse ■ Cah: C osine = adjacent hypotenuse ■ Toa: T angent = ○ opposite adjacent sin(x°) = cos(90°−x°) Radians ● Unit Circle ● Angles to Radians ■ Multiply by π 180° ■ Example: ● 270 °* ● Radians to Angles ■ Multiply by π 180° = 3π 2 180° π ■ Example: 27 ● 3π 2 * 180° π = 270 Complex Numbers ● i is a complex number that equals √− 1. ● i2 = − 1 ● i3 =− √− 1 or − i ● i4 = 1 ● Multiplying by the conjugate ○ ○ ○ ○ ○ ○ Example: 3+2i 4−i 3+2i 4+i 4−i * 4+i (3+2i)(4+i) 16−i2 12+11i+2i2 16+1 10+11i 17 28