Functions Flashcards
What are the first 7 rows of the Pascal’s triangle?
1
11
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
Describe the most basic forms of hyperbola, truncus, and square root graphs.
Hyperbola: exponential in 1st and 3rd quadrant
Truncus: symmetrical exponential at the same level
Square root: 90 degrees rotated parabola cut in half
What are the formulas that can be used to factorize x^3 expressions?
(x+y)^3= (x+y)(x^2-xy+y^2)
(x-y)^3= (x-y)(x^2+xy+y^2)
What is the pattern of cubic expansion?
First two always match +/- of original
Third ones always the opposite
Last ones always a +.
What is the formula to be used to find the amount of solutions and how do you use it?
b^2-4ac
>0 -> 2 solutions
=0 -> 1 solution
<0 -> no solution
What is the intercept form?
(x/a)+(y/b)=1
What does the formula m=tanθ calculate?
The angle a line makes with the horizontal.
What does the arrangement of simultaneous equation lines represent?
lines cross= 1 solution point
lines parallel, not crossing= no solution
lines overlapping forming 1 line= infinite number of solutions
What of the equation shows if lines will be parallel?
Ratio of x and y values- if ratios are the same then the gradients are equal therefore lines parallel
How to find value of m (in equation) that causes line to overlap/ be parallel?
use coefficients to form ratio
use cross multiplication to find 2 solutions of m
sub solutions back in to see if solutions are parallel or what.
What is the equation that can be used to find the axis of symmetry?
y= -b/2a
How to know what’s the point of inflection?
y= a(x+h)^3 +k
point: (-h, k)
The square root function exists in _______ only. (original without translation)
Quadrant 1
What is an even function?
When f(x)= f(-x)
What is a piecewise function?
Ones that has different rules for different parts of the domain.
Why is this equation not a function?
Because it failed the vertical line test.
What is the graph of an inverse function?
Line of f(x) reflected over line of y=x
What is the graph of an inverse function?
Line of f(x) reflected over line of y=x
What is the factorised form of quadratic equations?
y = a(x-e)(x-f)
expand (x+y)^2
x^2+2xy+y^2
What are odd functions?
f(x)= -f(-x)
rotational symmetry about the origin
e.g. y=x^3, y=1/x, y=sinx
At what circumstance can DRT be used for horizontal transformations?
When it’s factorized.
What to say when something is a function?
- each element in the domain has only one image
- passes the vertical line test
What to say when the inverse of something isn’t a function?
- F doesn’t pass the horizontal line test
- therefore can conclude that f^-1 fails the vertical line test
- therefore not a function
The student notices that with their translations the point (0. 5/2) is transformed to the point (1. 3/2). This confuses the student because they know that f(0)= 5/2 and f-1(5/2)=0
Explain why the translations do not work as the student expected.
An inverse function is reflected along the line 𝑦 = 𝑥 so that the image of a point
ends up on the other side of the line. therefore (0,5/2)-> (5/2,0)
When the points are translated right 1 and then down 1, they are not being
reflected along the line 𝑦 = 𝑥. therefore (0,5/2)-> (1,5/2)-> (1,3/2)
The point (1, 3/2) lies on both g(x) and f-1
(x)but is not the inverse image from f(x)