Basic Algebra Functions Graphs

Question 1

What are integers?

Answer 1

A whole number such as -4, 7, 1, 4, 100, -100, √25 (only works if it result in an full number)

Question 2

What are rational numbers?

Answer 2

A number that can be expressed in a simple fraction, a/b and b ≠ 0.

For example, -1.15 = -1.15/1, which is a rational number. 7 = 7/1, which is also a rational number. 0.41 = 41/100, which is also a rational number.

It is not a number that has infinite decimal expansion

Question 3

What are irrational numbers?

Answer 3

A number that cannot be expressed in a simple fraction such as π. √2 (since it will equal 1.41421….), Euler number (e) and numbers that have a bunch of random ongoing numbers.

Question 4

What are real numbers?

Answer 4

A number that can be in a number line or have an infinite decimal expansion, such as π, -2, 5, -7/8 etc.

Question 5

What are imaginary numbers?

Answer 5

The function for imaginary numbers is “ i “.

Anything you enter on a calculator that results in “ i “. Is an imaginary number.

It usually appears in a √of a negative number, such as √-2, √-7.

Question 6

What are | | symbols meaning?

Answer 6

It means how far the number is from 0 in a number line. For example, |-20| is 20 units far from 0 in a number line.

Question 7

What does this symbol mean >

Answer 7

Greater than

Question 8

What does this symbol mean <

Answer 8

Lesser than.

Question 9

What is a reciprocal?

Answer 9

A reciprocal is making a/b to b/a or x into 1/x or-y into 1/-y.

Most often, questions ask you to sum the reciprocal of the number, such as x + 1/x.

Question 10

What is the purpose of reciprocal?

Answer 10

This will simplify complex calculations. It will come in handy when dealing with questions that require adding numbers per something like km/time.

Question 11

What is the difference between an exact and an approximate?

Answer 11

Exact has no decimal places.
Approximate does.

Question 12

what is the rule when bracketing indices? such as (a/b)^x

Answer 12

The rule is, (a/b)^x = a^x/b^x.

For example, (x/3)^3 = x^3/3^3 = x^3/9.

Question 13

What happens when the power is ^0?

Answer 13

Equals to 1. 9^0 = 1.

5x^0= 5 (since 5 times 1 = 5)

Question 14

When both of the indices are in power of a negative number in a fraction, what do you do?

Answer 14

You minus both as usual. It will come as a positive number. The result exponent must be on the numerator.

Question 15

What do you do when you have a power that is a negative number?

Answer 15

When a power is a negative number, the rule is to take the reciprocal of the base (eg a^(-x) to 1/a…) and raise it to the positive value of the exponent (thus 1/a^(x). Basically flipping to make the exponent positive.

Question 16

What is a radical?

Answer 16

√ or commonly known as square root.

Question 17

What’s the purpose of a radical?

Answer 17

To find the “root” same number that resulted in the result.

Question 18

Why should you simplify a radical?

Answer 18

When an answer gives you an infinite decimal expansion, you must simplify it to get the exact answer (unless the question asks you to round up).

Question 19

How do you simplify a radical?

Answer 19

For example, √45.

Find the highest common factor that includes a squared number such as 4, 9, 16, 25, 36, 49, 64, 81, 100.

In this case, √45 = √9 times 5

9 can be radicalised so is 3

However, 5 can’t, thus is 3√5

Question 20

What is a polynomial?

Answer 20

When it has the leading coefficient, 2 different variables, integers and an operator.

Such as

5x^2y^2 + x^2 - 6.

A single number like 7 is a monomial.

Question 21

How do you deal with bracket power of 2 with polynomials?

Answer 21

If a^2 then (a)(a).

For example,

(10y-11)^2 = (10y-11)(10y-11).

Question 22

What if your multiplying many polynomials?

Answer 22

Follow BODMAS (Order from left to right)

For eg

(x+2)(x-5)(3x-2)

Start with (x+2)(x-5), then (3x-2).

It makes things way easier.

Question 23

What do you do when you’re multiplying two or more fractions?

Answer 23

Multiply across, all numerators together and all denominators together.

Question 24

What do you do when you’re dividing two or more fractions?

Answer 24

Keep Change Flip the 2nd, the 3rd, 4th whatever fraction.

Question 25

What do you do when you add or minus two or more fractions?

Answer 25

Make it a common denominator by multiplying each side of the other denominator which will result in multiplying the top. Then, you add the numerator.

Question 26

How do you factor a polynomial?

Answer 26

Pay attention. the format usually goes as x^2 +3x - 28. To factor a polynomial, you get the leading coefficient to multiply the last integer. You then find the HCF that multiplies the modified integer but also adds up to the middle coefficient. So in this case -28 can be multiplied by 7 times -4 and also result in the addition, of -3. So now is x^2 +7x - 4x - 28 Rationalise it x(x+7) - 4(x+7) Cross one of the same set and there you end up with (x+7)(x-4)

Question 27

How do you "Write the rational expression in the lowest term"?

Answer 27

You have to factorise the polynomial. For example r^3 + 64 The rule for factorising cube is (x+y)(x^2-xy+y^2) Factorised (r+4)(r^2-4r+16) Done. If a fraction, cross out the similarities as it will result in times 1.

Question 28

What are Quadrants places

Answer 28

If x and y = QI If x and -y = QIV If -x and -y = QIII If -x and y = QII

Question 29

What does it mean to “rationalise the denominator”

Answer 29

To make the denominator a natural number (eg 2,3,4) So do whatever it takes to just make the denominator clear

Question 30

What is a function?

Answer 30

Something that map a number and repalce the variable in an equation or formula. For example f(x) = 2/x add 4 then it would be f(4) = 2/4 = 0.5.

Question 31

What is the Domain of the function?

Answer 31

Set of all valid inputs over which the function has defined outputs. If it is not a defined output, is not part of the domain. Most likely, the output is a real number. The answer will look like this. if y greater than A if x is B and C. INPUT FOCUSED

Question 32

What is the Range of the function?

Answer 32

Set of all possible valid outputs. f(x) = x^2 outputs will be ≥ 0. (parabola). Can be the same as the domain OUTPUT FOCUSED

Question 33

How can you a perpendicular line, parallel or neither?

Answer 33

if m1 = m2 then is parallel if m1 * m2 = -1 then is perpendicular if neither then neither.

Question 34

How do you factor a polynomial that doesn't have 1 has the leading coefficient.

Answer 34

Once again, pay attention. 8x^2 + 10x - 7 = 0 Get the leading coefficient (8) and times the last number which is 7. You would get 56 Factor 2 numbers that multiples to -56 but also adds up to 10. 14x, -4x Add it 8x^2 + 14x - 4x -7 = 0 Now split it up, 8x^2 + 14x -4x -7. Factor each Two number common for 8 and 14? 2 2x(4x + 7x) Two numbers common for -4 and 7? 1 -1(4x + 7) overall 2x(4x+7)-1(4x+7) = 0 Cross the similarities and get the rest (4x+7)(2x-1) Now solve each 4x + 7 = 0, x = -7/4 2x-1 = 0, x = 1/2 THUS x= -7/4 , 1/2