Maths - Algebra & Graphs

Key Vocab: Variable, coefficient, constant, term, expression, exponent 

Like Terms: A like term is a term whose variables and exponents are the same. Note: the coefficients can be different. In Algebra, you can only add or subtract LIKE terms. Powers that are different values are NOT like terms. An example of a like term is 2x an x and an example of an unlike term is 4a and 4b.

Multiplying and Dividing: Multiplying and dividing can be done with like AND unlike terms in Algebra. The key steps are to... Multiply coefficients as you normally would. To show two variables are being multiplied, simply write them next to each other. If you have two of the same variables being multiplied, add the powers. If you have two of the same variables being divided, subtract the powers.

Powers and Exponents: When you have a power to a power (4x²)², multiply them and everything inside the bracket by what is outside. So this would be 16x⁴. 

Substitution: Simply switch the letter with the number and perform the required operation.

Inequalities: 

< less than       ≤ less than or equal to            > more than        ≥ more than or equal to

Solve as you normally would but flip the sign when you have a negative.

Rearranging: 

Keep performing the inverse operations until you are left with only the variable that you want on one side of the equals sign.

Key Words:

Add: plus, more than, greater than, sum, and, extra

Subtract: take away, remove, minus, less than, take off

Multiply: times, product, lots of, power, squared, cubed

Divide: divisible, share, goes into

Converting:

Percent to Decimal: Move the dp 2 places left and remove the percentage sign.

Decimal to Percent: Move the dp 2 places left and add the percentage sign.

Fraction to Decimal: Divide the numerator by the denominator.

Decimal to Fraction: Look at how many dp the decimal has. If it has two, make it out of 100.

Fraction to Percent: Divide the numerator by the denominator then multiply by 100 and add the percentage sign.

Percent to Fraction: Divide by 100 then use the steps to convert decimal to fraction.

ALWAYS SIMPLIFY

CRISS CROSS SMILEY FACE :)

Expanding:

One Bracket: Each term inside the bracket is multiplied by the term outside the bracket. Only multiply them by the number right next to the bracket. 

Two Brackets: Remember FOIL (First, Outside, Inside, Last)

Examples:

One Bracket: 3(x + 2) -> 3x + 6

Two Brackets: (x + 3)(x + 2) -> x² + 2x + 3x + 6 -> x² + 5x + 6

Special Equations:

Perfect Square: (x - 1)² -> (x - 1)(x - 1)

Difference of Two Squares: (x + 3)(x - 3)

Factorising:

One Bracket: Find the highest common factor between the terms in the expression. The common factor is placed outside the bracket. Inside the bracket goes the expression with each term divided by the common factor. 

Two Brackets: y = ax² + bx + c

To factorise you have to find two numbers that multiply to make c and add to make b. 

If c < 0, then one answer must be negative and one must be positive. If c > 0 and b < 0, then both of them are negative. If c > 0 and b > 0, then both of them positive. 

Special Equations:

If c is a sqaured number (even if it is negative) then you have either a perfect square or a difference of two squares. If b = 0, it is a difference of two squares. If not, it is a perfect square. 

Quadratics if a does not equal 1:

Take ax² x c and find two terms that multiply to make this and add to make bx. Replace bx with those two terms. Then split the equation in half and factorise each half into one bracket to get two brackets that are the same. Factorise the result to get a product of two brackets. 

Solving Quadratics:

Rearrange the equation so that it equals zero. Equate y (the quadratic) to zero. Then factorise the quadratic. Make each bracket equal to zero and solve the two equations. This will give you the two x intercepts of a parabola version of the equation.

Examples:

2 - 7x² - 24x - 9;    63x² - 24x;   -21x -3x;    -7x² -2x/-3x - 9;    -7x²(x + 3) -3(x + 3);    (-7x - 3)(x + 3)

y = x² - 4x - 21;     0 = x² -4x - 21;     0 = (x + 3)(x + 7);     x + 3 = 0 OR x - 7 = 0;      x = -3 OR x = 7

Linear Sequences:

A sequence is a list of numbers, in order, that follow a rule. A linear sequence is where the difference between consecutive numbers is constant (the same). A linear sequence has a plus or minus difference. An nth term rule gives the term in the nth position and can be used to find any value in the sequence. 

Linear Tables and Graphs:

When you enter sequences into a table by having the first row as the term. When you have the table, you can use it to plot the information by using the n values as the x axis. You can create a linear graph using three methods; 1) calculating the points 2) using the intercepts 3) using the general equation.

y = mx + c

y = mx + c

Intercepts: An intercept is a point where the line crosses the x or y axis. The general equation for a straight line is y = mx + c. In this equation, x and y are variables, c is the y intercept and m is the gradient.To calculate the y intercept we have to make x equal to 0. To calculate the x intercept we have to make y equal to 0.

Calculating the gradient (m): The gradient tells us how steep the line will be. The gradient is shown as a fraction that is change in y over change in x (also known as rise over run). 

Drawing the graph: Determine the y intercept by seeing what c (the number that is added/subtracted on the end) is. Determine the y intercept by seeing what m (the number before x) is. If it is a fraction, you have your rise over run equation ready. If it is a whole number, put 1 as the run.

Finding an equation from the graph: Determine the y intercept (c) by seeing where the graph intercepts y. Move along the graph until you find a point where it goes through an x and y coordinate precisely. Count how much the graph has moved up (rise) and how much it has moved across (run) to reach this new coordinate. Your gradient (m) is rise/run. Subsitute into y = mx + c.

Quadratic Patterns:

When you are given a pattern in a table and asked to find the rule, follow these steps. The rule will be formatted like ax² + bx + c. Find the first and second difference between the numbers in the second row. A = 1/2 second difference. Take the given pattern and take away the second difference multiplied by the term number squared. Find a linear pattern with the result of the previous step. Add the linear pattern to ax². 

Parabolas:

Using Intercepts: y = (x ± a)(x ± b)

To find x intercepts, make y = 0 and solve. To find y intercepts make x = 0 and solve. To find vertex find halfway between the x intercepts to find x coordinate and then substitute to find y.

Using the Vertex: y = (x ± a)² + b

The opposite of a is the x coordinate. B is the y coordinate.