Quartic graphs - Study guides, Revision notes & Summaries

Cubic A graph where the highest exponent is 3 Cubic function f(x) = ax³+bx²+cx+d Repeated root A function where at least 2 of the roots are equal. The turning point of the graph is on the x-axis E.g. y=(x-3)(x-2)² Point of inflection A function where all the roots are equal The graph flattens off on the x-axis E.g. y=(x-2)³ Sketching graphs with exponents Find the shape, roots, and y-intercept Mark any points of intersection with the axises - Shape is a...

area of sector (where θ is in radians) ½r²θ angles in 3D vectors cosθₓ =x/|a| cosθᵧ =y/|a| cosθ₂ =z/|a| this is where vector a makes an angle θ with the x/y/z axis (because x/y/z would be the adjacent and a the hypotenuse) Expanding (a + bx)ⁿ (a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ cosine rule a² = b² +c² -2bc cosA rule for parametric integration ∫y dx = ∫(y dx/dt)dt as (dx/dt) x dt would cancel out to give you ∫y dx un...

Maths A level Questions and Answers Rated A What is 2^x multiplied by 3^x? Let x=2 2^2 X 3^2 = 4 X 9 = 36 36=6^2 So answer is 6^x How would you solve (x^3 - 2x)/3x^2 Break down fraction to (x^3/ 3x^2) - (2x/3x^2) then continue to simplify fractions and cancel out like terms How would you factorise 4x^4 - 13x^2 + 9 Change 4x^4 and 13x^2 into 4(x^2)^2 and 13(x^2) Then let (x^2) be y and substitute y to get 4y^2-13y+9=0 which factorises into (4y-9)(y-1)=0 Which is equal to (4x^2 -9)(x^2 -1) This ...

Gradient y2-y1/x2-x1 Length of a line sqrt((x2-x1)^2+(y2-y1)^2) Midpoint (x1+x2/2, y1+y2/2) How do you know if two lines are perpendicular? When you multiply the gradients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle How do you know if two lines are parallel? The gradients should be the same What should you always remember when finding the square roots? The answer could...

Pure Maths A Level Questions and Answers 100% Pass Gradient y2-y1/x2-x1 Length of a line sqrt((x2-x1)^2+(y2-y1)^2) Midpoint (x1+x2/2, y1+y2/2) How do you know if two lines are perpendicular? When you multiply the gradients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle How do you know if two lines are parallel? The gradients should be the same What should you always remember when finding the square roots? T...

1. Gradient y2-y1/x2-x1 2. Length of a line sqrt((x2-x1)^2+(y2-y1)^2) 3. Midpoint (x1+x2/2, y1+y2/2) 4. How do you know if two lines are perpendic- ular? When you multiply the gra- dients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle 5. How do you know if two lines are parallel? The gradients should be the same 6. What should you always remember when finding the square roots? The answer cou...

1. Gradient y2-y1/x2-x1 2. Length of a line sqrt((x2-x1)^2+(y2-y1)^2) 3. Midpoint (x1+x2/2, y1+y2/2) 4. How do you know if two lines are perpendic- ular? When you multiply the gra- dients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle 5. How do you know if two lines are parallel? The gradients should be the same 6. What should you always remember when finding the square roots? The answer cou...

Unlock the power of visual learning with our meticulously crafted digital A-Level Maths Graphs Notes, designed to elevate your understanding. Perfect for A-Level students and educators alike. The notes covers a spectrum of graph types, including cubic, quartic and reciprocal graphs. There are many resources available online however, it is time consuming to look through them all. I have gone through countless textbooks, YouTube videos and exam questions consolidating all the knowledge into one...

This document introduces polynomial and power functions, categorizing them by degree from linear (1) to quintic (5). It outlines the general forms of each type, emphasizing how polynomial functions change based on their degree and the leading coefficient's sign. For each degree—linear, quadratic, cubic, and quartic—it includes the function's classification, graph behavior, domain (possible x-values), and range (possible y-values). Linear and cubic functions have unrestricted domains an...

Exam (elaborations) TEST BANK FOR Calculus 8th Editon (3 Volume Set) By Ron Larson, Robert P. Hostetler and Bruce H. Edwards (Study And Solutions Guide) C H A P T E R P Preparation for Calculus Section P.1 Graphs and Models . . . . . . . . . . . . . . . . . . . . . . 2 Section P.2 Linear Models and Rates of Change . . . . . . . . . . . . . 7 Section P.3 Functions and Their Graphs . . . . . . . . . . . . . . . . . 14 Section P.4 Fitting Models to Data . . . . . . . . . . . . . . . . . . . ...