Subsets of Real Numbers

1. KR 1 • Real Numbers (R): All numbers are real. {3.876, 3.-9, 7, 0, ½, 10} • Irrational Numbers (I): The classification of irrational numbers consists of numbers that do not have fractions or integers, the are usually numbers that are a decimals that go on forever and do not have a specific pattern. {, 3.14…} • Rational Numbers (Q): All numbers including decimals that contain a pattern. All numbers except for irrational. {3.876, 3., -9, 7, 0, ½, 10} • Integers (Z): These numbers are all numbers that are not fractions or decimals and can be both positive, negative, and zero. {-9, 0, 7, 10} • Whole (W): Numbers that cannot be a negative, decimal, or fraction. They can be zero and can be positive. {0, 7, 10} • Natural (N): These are numbers used to count. They do not have negatives, fractions, decimals, or zero. {7, 10} Subsets of Real Numbers

2. 2 F.H. Properties of Real Numbers

3. To start off you need to get the and X by itself so you would divide by 3 Then you would make a line following this equation on a number line(since it’s 1 variable) or a graph • The circle is open because it is not equal to • The thing with the () is called interval notation. • To solve a compound inequality. • Get Y alone by diving by – this also flips the signs • Make this into 2 inequalities Example: 3x< 9 3 X<3 (3,infinity) Example: -3X<-Y<8 3X>y>-8 3x<y and y>-8 Graph both of these and were they intersect is all the possible answers. The flat line shades upwards Since it is > and the other is Also up since it is > The lines are dashed cause They cannot be equal to. Solving and graphing simple and compound inequalities To find all the possible X values that make the inequality true

4. #7 Graphing a line Step on How to Graph a Line: Ex: Identify the Variables- are they independent, which is the first set of data, or the dependent, which is changing because of independent Label each axis Determine the variable range- subtract lowest value from highest and that is the range. Number each axis Plot the data points Draw the line (it is best to use a ruler) Title the graph Make sure the lines are not curved, something may have gone wrong First I got Y by itself. Then I had to get the number in front of it gone. I divided 2 by 10 and 3. Then I got Y= 3x-5. I started at point -5. I went up 3 and over to the left one. Then I went down three and over to the right one.

5. #8 Determining a Function Equation = If the y has an exponent it is not a functionSet of Points = if the x repeats it is not a function Graph- if it passes the vertical line test • Examples of Functions: • y=3x+2 • (1,2)(3,4)(8,9) • Y= 34x-4 A function relates and input to an output X = 5 X= -

6. 9 VK The equation of two points can be solved by using the point-slope formula [y-y1=m(x-x1).] Finding the Equation of Two Points Explanation: (1,8)(2,6)- Example m= 2/-1 (-2). First find the slope, this is the m in the formula and rise over run. y-y1=m(x-x1) y-6=-2(x-2). Choose a x value and a y value from one point, I chose y=6 and x=2. y= -2x+10 The actual formula was found by replacing y1with a y value from one of the points and the x1 was replaced by the x value from the same point. After the points were put into slope-intercept formula (y=mx+b) graph the points using y= -2x+10and graph the equation.

7. 10 JS Example: GIVE AN EQUATION PARALLEL TO X=2 THROUGH (4,5) FIRST YOU WOULD NEED TO KNOW THAT X=2 CAN BE GRAPHED VERTICALLY As  NOW WE NEED TO FIGURE OUT THAT A VERTICAL LINE WOULD BE PARALLEL TO A VERTICAL LINE. SINCE ONLY X=4 IS A VERTICAL LINE THAT MAKES IT PARALLEL TO X=2 Finding Perpendicular and Parallel lines through equations Example: GIVE AN EQUATION PERPENDICULAR TO X=2 THROUGH (4,5) FIRST YOU WOULD NEED TO KNOW THAT X=2 CAN BE GRAPHED VERTICALLY As  NOW WE NEED TO FIGURE OUT THAT A HORIZONTAL LINE WOULD BE PERPENDICULAR TO A VERTICAL LINE. SINCE ONLY Y=5 IS HORIZONTAL THAT MAKES IT PERPENDICULAR TO X=2 THAT WOULD BE THE ANSWER.

8. #11 Equations for Parallel and Perpendicular lines Perpendicular Lines Ex: Parallel Lines Ex: Find an equation to the line perpendicular to y=-3 though (-2,5). Solution: Since x is already by itself, we proceed to solving for the passing though line (when answering a graphing question you would plot the x on the x axis.) We use the y-y1=m(x-x1) formula to solve. M stands for slope and x and y stand for the axis's. The answer would be y=5. Also when looking at the insect (-2,5), the y number is 5. The answer will not always be like that though. Find an equation of the line parallel to the line y=5 and passing though (-2,-3). Solution: Since the graph of y=5 is a horizontal line, any line parallel to it is also horizontal. The equation of a horizontal line can be written in the form of y=c. An equation for the horizontal line passing though (-2,-3) is y=-3.

9. Finding Equations given: Parallel/Perpendicular through a point to a vertical/horizontal line through a point. • Ex: Find an equation of the line parallel to the line y=5 and passing through (-2,-3) • y=-3 Equations through lines

10. 13 SO By: Serene Ong Domain: Set of x-values that will “work.” Range: Set of y-values that can be received. Example 3 Function: y=x^2 Domain: So you can plug anything in since all values will be non-negative. Range: [0, ) To find the range, you must use a graph. Finding Domain and Range in Interval Notation Example 1 Function: f(x)=x-1 Solve for what x can/can’t equal to get… Domain: All real numbers, since you can plug in anything . Example graph: Domain: Range: Example 2 Function: g(x)= Solve for what x can/can’t equal to get… Domain: therefore making ). This means that any number except one will “work.” Range Domain

11. #14 On an equation, you have to get X and Y by itself and the other variable equals zero. Ex: 4y-x=3 • Set y equal to zero and get x by itself: x=-3 • The x intercept is (-3,0) • Set x equal to zero and get y by itself: y=3/4 • The y intercept is (0,3/4) X and Y intercepts

12. 16 TS Solving Systems using Graphing Get both equation into the format y=mx(Slope)+b(begin) -6x-3y=-27 3x+2y=16 ---------------- Y=-2x+9 Y=-3/2x+8 ---------------- Now graph both and where the lines intercept is the answer. (2,5) 3. Do not forget to check your answer,

13. By Ben Ferraro • Function Notation Function Notation is really just the notation f(x) "f of x" This can be used in a function to show what y equals, given an x value. A number or variable can be substituted for the x and then can find the y value once solved. This can also be used to solve multiple equations and variables can be combined like f(g(x)). You would solve for the g then plug in the answer of g into f's to solve that equation as well. • Examples: • f(x)=2x+1 plug in 3, f(3)=2(3)+1 the y value is 7 and the coordinates would be (3,7) • f(x)=2x+1 plug in b, f(b)=2(b)+1 • f(x)=2x-4 and g(x)=5x+1 use f(g(2)) g(2)=5(2)+1=11 then use f(11) f(11)=2(11)-4= 18

14. Substitution is a way to solve a system where you get one variable by itself. If the equation was You have to plug in the x from problem number two into problem number one. Get the x by itself. X=-4y+18 and then plug it into the first equation. 2(-4y+18) +3y =16 -8 + 36 +3y =16 28 + 3y =16 then subtract 28 over which is 3y= -12 and divide by 3. y = 4 Plug in the x in the second equation x+ 4(4)=18 x+16+18 subtract 16 over. X= 2 The solution is written in coordinates (2,4). This is where the lines intercept. This is another example Solve a System of Linear Equations by Substitution A.P

15. Chapter 4, Solving by Elimination L.D 1. Start with the equation and decide which variable you want to eliminate. 2. Multiply the equation you would like to change to make the coefficient of the variable you chose match the other equation. 3.If the variables are both positive or both negative change one line to the opposite. (be sure to change EVERYTHING!) 4. Now you add the two lines together, and solve for the variable you didn’t chose to eliminate. 18y=18 y=1 x-4y=-2 x-4(1)=-2 x-4=-2 x=2 5. Plug in the answer you just got, to solve for the variable you eliminated. 2x+10y=14 2(2)+10(1)=14 4+10=14 14=14 6. Check your work! x-4y=-2 2-4(1)=-2 -2=-2 7. Final answer! This is where the two lines will cross. (x,y) (2,1)

16. #20 Graph: y > -x-2 y +5x < 2 Follow the equation y=mx+b 1. The first inequality is already in correct format so we leave that alone. • In the second inequality, you want to get y by itself: y<-5x+2 3. Graph: A. The dashed line represents a > or < sign B. The solid line represents a ≥ or ≤ sign C. The shaded area repesents all the possible points to make the two inequalities true. Systems of Liner Inequalities

17. These problems can be used to find the rate, time, and distance of two people or two moving objects. • Sample question: Bob and Jim live on two farms four miles apart. They decide to meet each other in between their farms. They meet after 1 hour. If Bob can walk two miles per hour faster than Jim, find how fast each of them walks. • Step #1: Let statement • The let statement represents the rate of an object • The let statement should look like this… Let x= Bob’s rate y= Jim’s rate • Step #2: Set up the equation • 1st part: 1x+1y=4 • The ones in this equation represent the time it took for them to meet; the four represents the distance they had to cross. • 2nd part: X=y+2 • In this equation, Bob(X) can walk two miles per hour faster than Jim(Y) so X=Y+2. • 3rd part: put together the system • The equations are put into one system that represents both equations • This system looks like this… • Step #3: Solve • the system is this… • 1st part: We know the value of X so we can plug that value into the top equation 1x+1y=4. • We then get 1(y+2)+1y=4 after we plug in the value of X • 2nd part: solve the top equation 1(y+2)+1y=4 to y+2+1y=4 after multiplying 1 across to the numbers in parentheses • We then subtract 2 from each side and add 1Y and y to get 2y=2 • We then divide both sides by 2 to get y=1 • 3rd part: plug Y=1 into x=y+2 to replace the Y value with 1 to get X=1+2 • Then you simplify and combine like terms to get x= 3 • The X and Y values represent the rates of Bob and Jim therefore the answer to the problem is Bob walks 3 miles per hour and Jim walks 1 mile per hour. Word Problems Using two Variables: Distance

18. These equations are used to figure out how much of something needs to be used to get a certain percentage value of something else. • When the two values being added are less than the desired end product, the equation cannot be completed. • Sample question: The chemist needs to mix 20% acidic solution with 30% solvent to get 2 liters of 25% acid solution. How much of each solution should he mix to get the 25% solution • Step #1: Let statement. • The let statement MUST have two variables (X and Y) • It looks like this… Let X= Amount 20% solution(Lower percentage number) Y= Amount 30% solution(Larger percentage number) • Step #2: Put together the equation • Make the equations • X+Y=2 • 0.2x+0.3y=0.25(2) • The system should be set up like this… • Step #3:Solve the equation • Addition method • Multiply the bottom numbers by 100 to create whole numbers • Multiply the top numbers by either -20 or -30 to eliminate one value to • Eliminate numbers Add -20y and 30y to get 10y and -40 and 50 to get 10. The resulting equation is… 10y=10 • Next divide by 10 to get Y=1 • You then plug 1 into the equation x+y=2 and subtract over the 1 to get x=1 • The X and Y values are the amounts of each solution that should be added togetherso the answer to the problem is 1 liter of 20% solution should be added to 1 liter of 30% solution to get two liters of 25% solution. • Decimal values from percentages Amount of solution needed Word Problems Using two Variables: Mixtures

19. Rule Example 1. When dividing numbers with the same base, subtract the exponents 2. When multiplying numbers with the same base, add the exponents 3. When simplifying negative exponents, flip the exponent to make it positive 4. When you have an exponent raised to an exponent, multiply the expression 5. When the exponent is to the power of 0, the answer is always 1 6. When the exponent is to the power of 1, the answer is always the base • When dividing numbers with a different base, add the exponent to each base. 8. When dividing numbers with a different base and a negative exponent, switch the denominator and numerator to the opposite side to make the exponent positive. 1. 6^3 = 6^3-2 = 6^1 = 6 6^2 2. 2^2  2^1 = 2^3 = 8 3. 4^-2 = 1 = 1/16 4^2 4. (2^3)^2 = 2^32 = 2^6 = 64 5. 3^0 = 1 6. 4^1 = 4 7. (8/4)^2 = 8^2/4^2 = 64/16 = 4 8. (8/4)^-2 = 4^2/8^2 = 16/64 = 1/4 Exponent Rules

20. Why? Let’s think about it… • If an expression was , we would add the exponents and get . But what if we didn’t subtract the exponents? is 9, and is 1/9. If we multiply them together (*), we get 1. • . So, to make a negative exponent a positive, move the exponent and the number it is attached to to the other side of the fraction. • Example: • The answer is . and

21. Solve: Kirti Dasari 3.72 = A1 3,720,000,000 3 Since the exponent is positive you move the decimal to the right. 567 = 0.00567 Since the exponent is negative you move the decimal over to the left. (Hint: Start at the five then move to the left.) (2.1 )(1.6 ) = 3.36 First, multiply the 2.1 and the 1.6. Then, add the exponents together. 52.4 = 5.24 Now, this may look right, but it’s not. You have to move the decimal so it’s right after the first number. Then you must change the exponent to match the number of times it has been moved over. 3.4104 + 5.2 = 5.54 When adding you must have the same power of 10 on both numbers. You can rewrite either of the numbers to match the power of 10. I am going to rewrite 3.4 . First, I am going to move the decimal over and multiply it by Like so, (0.34 ) 5.2 . After doing that your problem should end up like this, 0.34 + 5.2 . Lastly, you add the 0.34 and the 5.2 getting you to your answer. You do the same thing when subtracting. Instead of adding the number you subtract. Remember, when subtracting and adding you must make sure that they have the same power of 10. Scientific Notation

22. Long division Then you repeat: Distribute Subtract again And Now You're Done - Say your problem is1styou Make sure to Then you Distribute And you get -after you Distribute: After subtracting bring down the next term Note: Say you say you have a remainder instead of 0 at the bottom, put it over the divisor. For example, you remainder is 3, your answer is X-5 + 3/X+2.

23. Step 1: The x in the denominator has to equal 0 so 2-2=0. Put +2 in the front. +2 1 -2 3 -4 Step 2: Bring down the first number. Then multiply 2 by the number you brought down. Step 3: Add that column together. Take the answer from that second column and add it to the next number in the third column. Take that answer and multiply it by the number you brought out in the beginning. +2 1 -2 3 -4 +2 0 6 1 0 3 2 Step 4: The last number is the remainder. The third number is the constant. The second is the x and the first is the Then put the remainder on top of the original denominator. Answer: Synthetic Division(Ch. 5)

24. 27 DR Products • (a + b)2 = a2 + 2ab + b2 You need to do a2first. Then you need to do a * b * 2. Then you need to do +b2. It can’t be negative b2. This makes: a2 + 2ab + b2. If it was (a – b)2, then the final output would be a2 – 2ab + b2. This is called square-multiply-square. • (a + b)(a – b) = a2 – b2 To do this, you need to first take a2. Then you need to –b2. The final output will be a2 - b2. These are called conjugates. • (a + b)(a + c) = a2 + ac + ab + bc This is called double distributing, but you can use FOIL. First, outer, inner, last. That is the order in which you multiply. So it would be (F)a2 + (O)ac + (I)ab + (L)bc. (a + b)2, (a + b)(a – b), (a + b)(a + c).

25. American Academy Algebra I 2013

26. TWO TERMS: Difference of squares; Sum/Square of cubes • THREE TERMS: Perfect Square Trinomial; “Regular” Factoring • FOUR TERMS: Grouping • FACTORING LIST: • 1) Proper Order=0 • 2)GCF Lead Coefficient (-)=> 1 • 3)Two Terms:Three Terms:Four Terms: • #1Difference of squares #1 Perfect square trinomial #1 Grouping • a - b a +2ab+b x +3x-4x-12=0 • (a-b)(a+b) 2xa x b (x +3x)+(-4x-12)=0 • (a+b) x (x+3)+-4(x+3)=0 • (x-4)(x+3)=0 • (x+2)(x-2)(x+3)=0 • x=+-2,-3 • EX 1:EX 2:EX 3: • 25m -16 =0 25x -40x+16=0 15n -12n +10n-8=0 • (5m-4)(5m+4) 2*5x * 4 3n (5n-4)+2(5n-4)=0 • m=+-4/5 (5x-4) =0 (3n +2)(5n-4)=0 • x=-2/3, 4/5 By: Julian Burd 2 2 2 2 3 3 3 #2 Sum/difference of cubes 3 a +-b 2 2 #2 “Regular” (a+-b)(a +-ab+b ) 2 2 Examples: 2 Difference of squares #2 “Regular” FACTORING AND SOLVING 2 2 3 2 x=4/5 2 2 2

27. Simplifying Radicals (Ch. 8) Step 1: Prime Factorization: Separate the fraction into two radicals. 54 – 27 – 9 - 3 = | | | | Step 2: Simplify the radicals. 2 3 3 3 2X3=6 so the 6 goes inside Answer: 3because it is not a square number. Step 1: 54 Factor out 54 so that one of the numbers is a perfect square. Step 2: The number that is the perfect square goes on the outside of the radical. The number that is not a perfect square goes inside the radical. Step 3: Reduce the index. Answer: 3

28. 34 JS When we rationalize, it is mainly because it is easier to see the problem without a radical in the denominator. So we move it to the numerator by rationalizing. In calculus, the radical is left in the denominator and ignored. Rationalizing Problem to rationalize: There should never be a radical number in the denominator. So what you do is you multiply both the numerator and the denominator by your radical When you multiply a radical by a radical number the radicals cancel out and you just do regular multiplication from there. Problem to rationalize: Since we have a real number in the denominator, we cant just multiply by the radical. So we then multiply it by its conjugate. We then get because we distribute the 1 in the numerator = But you can go even further because 4 – 2 simplifies

29. 37 CB • Exponents that are fractions. • (n = the nth root of A) • So Examples: • = • = = 6 • = = 2 = = • (= 6) = = 7) = Fractional 2 1 Before we couldn’t simplify these problems further, due to different indices. Now we can switch to exponents, which can be simplified further. 2 Rational Exponents 6

30. Radicals with Operations Example

31. Page 29, B.O. Rational expressions are polynomials that have to be written as a quotient (fraction). Restrictions also have to be stated in the answer to inform the student that the denominator cannot equal zero (0) because if the denominator equals zero (0), then the expression will be classified as undefined. Example 1: (4-x2)/(3x2-5x-2) 1.We have to factor this expression. -1 (x-2)(x+2)/(x-2)(3x+1) 2. We have to simplify. (x-2) in the numerator and (x-2) in the denominator cancel out. Next, distribute -1 to (x+2) which will transform to -2-x. The final answer is: (-2-x)/(3x+1) Restrictions: x = -1/3 Example 2: (5m)/(2n) + (m)/(2n) Example 3: (1/2 + x)/(1/3 + 1/x) 1.We have to add the numerators. 1. We have to make a common denominators. (5m+m)/(2n) (1/3 + 1/x) = (x+3)/(3x) 2. We have to simplify. 2.We have to copy-dot-flop. (6m)/(2n) = (3m)/(n) (1/2 + x)/(x+3)/(3x) = (1/2 + x) (1/3x3 + 4/3x2 + 1/2x) 3. We have to distribute. The final answer is: (3m)/(n) Restrictions: x = 0 The final answer is: (1/3x4+ 3/2x3 + 7/6x2 + 1/4x) Chapter 7.1: Simplifying Rational Expressions

32. To solve rational equation, you have to get x by itself. Example: 1. 1/x+2=4 2. 1/(x+2)+3=2 Rational Equations Explanation: 1/x+2=4. First, there cannot be a denominator and the only way to get rid of the denominator is to multiply the whole equation by the denominator. (multiply by x.) The result after multiplying by x should be 1+2x=4x. Finally, get the x to one side by subtracting over 2x and dividing by 2. The final answer should be x=1/2. 1/(x+2)+3=2. Again, get rid of the denominator by multiplying the equation by the denominator, (x+2) and the result should be 1+3x+6=2x+4. After subtracting 4, adding 1, and subtracting 3x over, the answer should now be x=-3.

33. Flip bottom fraction • Move bottom faction to the top fraction • Multiply both fractions • Simplify if needed EX. • 5/6 * 10y/11y • 50y/66y • 25/33 Solving Complex Fractions

34. EX: It takes the 1st person 5 hours to paint a house. It takes the 2nd person 3 hours to paint the same house. How long will it take to paint the house if both of them work together? Let x = time If the 1st person can do the job in 5 hours, then in one hour he can do 1/5 of the job. If the 2nd person can do the job in 3 hours, then in one hour he can do 1/3 of the job. So together, if it takes x hours to do the whole job together, then we can make in equation like 1/5+ 1/3= 1/x. Find LCD(Least common denominator) which is 15 since 5x3 is 15 15(1/5 + 1/3) = 15(1/x) 3 + 5 = 15x then add the like termswhich equals 8 = 15x x = 15/8 and that can be reduced to 1 7/8 hours whichis 1.875 whichisabout 1.9. So multiply 1.9x6 so itwould be 1 hour 54 minutes. A.P Time Word Problems

35. #39 Pythagorean Theorem Word Problems 57,600 = 44,800 A surveyor must determine the distance across a lake. If C is 320 feet and A is 240 feet then find B. Subtract 57,600 from both sides C =320 ft. B A =240 ft.

36. A midpoint is a point in a line segment or an arc that is in the exact middle. • Ex: (6,4);(2,8)(add both x values and add both y values.) • (4,6) • Distance between two points:This can be solved by finding both distances and then using the Pythagorean theorem to find the other side . • Ex: Two points (4,4) (3,2) Subtract x values and y values. Then make label them as sides on the triangle. • c= • Then solve using the Pythagorean theorem. • Example: Steve went north for 2hrs. traveling 15mph, and Joe went west for 2hrs. traveling 14mph. After two hours how far were they apart? 40 JS 2 1 Steve Answer: They are approximately 41 miles apart 30mi 15mphx 2h = 30 mi 17mph x 2h = 34 mi Midpoints/Distance Between Two Points 784 + 900 = c2 1684= c2 41c a2 + b2 = c2 Joe distance=a Steve distance=b 282 + 302 = c Joe 28 mi

37. 41 P.W. • The Cycle: • i1 = √(-1) • i2 = -1 • i3 = -i • i4 = 1 Examples: i25 =(25/4= R1)= i i26 =(26/4= R2)=-1 i27 =(27/4= R3)=-i i28 =(28/4=R0)= 1 (9*-1) (2+3i)2= 4+12i+9i2= 4+12i-9= -5+12i *= multiply /=divide R= Remainder Steps: Simplify the problem. Find your imaginary numbers (marked with ix). If the imaginary number is just i, leave it. If not find the exponent on the cycle and multiply the coefficient by the number in the cycle. If the exponent is not on the cycle then divide it by four ONCE, then if it is perfectly divisible then the ix should just turn to a 1 and leave it. If it is not then take the remainder and make that the exponent and then follow step 3. (If possible) Simplify the problem. IMAGINARY NUMBERS

38. 42 K.M. √ When solving by square root, remember that you must have two answers: Positive (+) and Negative(-) Solve By Square Root Ex:2 Get the square alone, or the parentheses with the square alone, and then square root both sides of the equal sign. Make sure to divide the coefficient out before you take the square. There are always two answers, because of the square root. Examples: c) c) c)

39. 43- MD Without a Lead Coefficient: +16 +16 = x+4= X=-4 X= -2, -6 Completing the Square With a Lead Coefficient: Definition: +bx )- - =9/4 X-1= X=5/2, -1/2 +1 +1 X 1

40. Solving Polynomials by the Quadratic Formula; Discriminating O.L.L 44 The quadratic formula is: This is read “the opposite of b plus or minus the square root of b squared minus four ac all over two a.” Here is the problem: x2+3x-7. Here is discriminating: Here is how to solve using the quadratic formula with this problem: x2-3x-10 If there are terms missing, their coordinating letter will always equal zero. If the square root is a negative, then just put an “i” in the answer(s).

41. Find the Axis of Symmetry (AOS) by using the formula X=-b/2a • Find the vertex by plugging in the AOS as X (AOS, after plug in) • Next use the C digit or the one without the variable as your Y- intercept. Make sure X=0 • Next use the square root rule, completing the square (CTS), quadratic equation, or factoring to find the X- intercept. Make sure Y=0 • Next if you do not have five points on your graph then Plot • Finally graph • IMPORTANT: make it so the lines on the graph don’t look like a V, don’t go sideways, and don’t curve in. EX. X2+4x+4=0 1. -4/2(1) -> -4/2 -> -2 2. (-2,0) -> (-2)2+4(-2)+4 -> 8-8=0 3. (0,C) -> C=4 -> (0,4) 4. (CTS) X2+4x+4=0 -> (X2+4)=-4 -> (x+2)2=0 -> x+2=0 -> x=-2 -> (-2,0) 5. Plot X=-1 I Y=3 -> (-1,3) Graph Graphing a Quadratic

42. Vertex Form 47 SO Standard form: y = ax2  + bx + c Vertex form: A(x-h)2 +k V(h,k) By: Serene Ong How to graph from vertex form Example of vertex form: f(x) = (x-2)2+ 4 How to solve from standard form to vertex form: Axis of Symmetry=2 V(2,4) Y-intercept=(0,8) X-intercept=Imaginary(doesn’t touch x axis) -4 +4 y=x 2 -4x+8 y=(x2 2 -4x)+4 (x-2) 2+4 To find the vertex, you also take the number inside the parentheses and make it the opposite. Once you have that number, take the constant being added and that’s your y coordinate. To find the axis of symmetry, take the number inside the parentheses and make it the opposite. If it’s a positive, make it a negative and vise-versa