Polynomials and Quadratics Flashcards
shape of graph of a quadratic
parabola (smile=minTP, frown=maxTP)
quadratic formula
x=-b+/-square root of b squared - 4ac all over 2a
discriminant formula
b squared - 4ac
completing the square
in the form y=a(x+p)squared +q axis of symmetry x=-p turning point (-p,q) *can check answer by expanding brackets*
steps of sketching parabola
- find discriminant
- cuts y-axis, x=0
- cuts x-axis, y=0
- calculate axis of symmetry/minTP
determining equation of parabola
y=k(x-a)(x-b)
- work out roots
- factors
- sub point in to work out k
- write equation
solving quadratic inequalities
need to know shape (max or min) and roots
sketch graph
write out answer
steps on how to prove tangency and find point of contact
- equal equations to each other
- equal to zero
- factorise and find roots (should be repeated root)
- sub x coordinate (root) into one of the equations to find y coordinate
steps on how to find equation of tangent
- sub gradient into y=mx+c
- equal equations to each other
- equal to zero
- discriminant is equal to zero, find c
- find equation using y=mx+c
degree of a polynomial
value of the highest power
synthetic definition steps
- write coefficients across top (put zero if there isn’t one)
- write number you want to evaluate at left
- bring down first coefficient
- multiply by side number, add onto coefficients
- number at end is remainder
sentence at end of synthetic division
since remainder is zero, x-4 is a factor and x=4 is a root
factorising polynomials
use factor found in synthetic division and numbers on the bottom of synthetic division
factorise or repeat synthetic division if necessary
use number at end of polynomial to find factors
using synthetic division to solve equations
exact same steps as before but find roots of the factors at the end (equal them to zero)
quotient and remainder
quotient are the numbers at bottom of synthetic division and remainder is number at end
finding unknown coefficients
- use synthetic division
- equal remainder to zero
- rearrange to find unknown
finding multiple unknown coefficients
exactly the same as before but use simultaneous equations at the end.
determining equation of a curve (when roots are known and at least one other point)
- right out roots
- rearrange to get factors
- write out y=k(x-a)(x-b) with factors
- sub in known point to find k
- write out final equation
* if repeated root, write twice in equation*
what is a polynomial?
an expression containing the sum or difference of algebraic terms with powers or the equivalent in factorised form.
when trying a divisor of the form (x-a) it is usually a good idea to start with…
(x-1)
how to determine the equation of a polynomial from its graph
- use the roots to determine factors eg roots a,b,c give factors (x-a)(x-b)(x-c)
- remember the polynomial may have a scalar, k eg y=k(x-a)(x-b)(x-c)
- substitute coordinates of y-intercept to determine k.
how to determine points of intersection of line and polynomials
- make the equations equal and rearrange to equal 0.
- factorise using synthetic division
- identify and interpret roots –> equal roots are required for tangency
- find any other points of intersection
completing the square is used to:
- determine the coordinates of the turning point
- solve a quadratic equation which:
- does not factorise
- is undefined using the quadratic formula
- has discriminant<0, no real roots
- gives a parabola that lies entirely above or below x-axis.
what form does a quadratic need to be written in to complete the square?
ax^2+bx+C
quadratic equations can be solved by:
- using the graph
- factorising
- using quadratic formula
- completing the square
how to solve quadratic inequalities
- solve the related quadratic equation to find the roots
- make a sketch of the graph of the quadratic function
- identify the solution of the inequality:
f(x)<0 has solutions below x-axis
f(x)>0 has solutions above x-axis
if b squared -4ac<0…
no real roots
if b squared -4ac=0…
real and equal roots
if b squared -4ac>0
real and distinct roots
how to determine relationship between a line and a parabola
- make equations equal and rearrange to 0
- find and interpret the discriminant of the resulting quadratic equation
=0 tangent
>0 crosses parabola in 2 places
<0 does not touch parabola - find the coordinates of any points of contact
