easy math memorisation for exam sem 1

discriminant formula

Δ=b2−4ac

Δ>0

Two real solutions (roots), the parabola intersects the x-axis at two points.

The range for m must lie between the roots so <0 if -ve and >0 if +ve

Δ=0

One real solution (root), the parabola touches the x-axis at exactly one point (vertex).

Δ<0

No real solutions (roots), the parabola does not intersect the x-axis.

The range for m must be greater than 0 for no real solutions, as it positions the parabola above or below the x-axis, preventing any intersections and making the discriminant negative.

When do you use the discriminant formula?

analyze the number of real solutions (roots) of a quadratic equation

whether the parabola intersects the x-axis, touches it, or doesn't touch it

irrational root vs rational root

perfect square solutions vs solutions containing surds

unknown values other than ‘x’ role

Controls the vertical position of the parabola, affecting the vertex location and determining whether the parabola intersects, touches, or misses the x-axis, thus influencing the number of real solutions.

polynomials

have only positive whole number powers for the value of x with coefficients

What is function notation?

f(x) represents the output of a function (y) for input x. it can only be used for polynomials

How do you evaluate f(x)?

Substitute the given x-value into f(x)Example: f(2)=2(2)+1=5

What is the domain and range?

Domain is all possible input values x, and range is all possible output values f(x)

*What is f^−1(x)

f−1(x) is the inverse function, reversing the input-output roles of f(x)

What is function composition?

How to find the inverse of f(x)=2x+3

Swap f(x) and x, solve for y: f^−1(x)=x−3/2

What is the Factor Theorem?

If f(a)=0, then (x−a) is a factor of f(x).

What does the Factor Theorem find?

It finds the roots (x-intercepts) of a polynomial.

How to use the Factor Theorem?

Set f(x), find values of x, and factor the polynomial.

Example: Find roots of f(x)=x^2−5x+6

Factor: (x−2)(x−3)=0 Roots: x=2, 3

What are x-intercepts?

Points where f(x)=0f(x) = 0f(x)=0, the graph crosses the x-axis.

when to equate the coefficients (using a given factor)

A factor of a polynomial is (x−r) if f(r)=0 where r is a root of the polynomial.

factor theorem (equating the coefficient)

to find the roots

The order of transformations

The order to perform transformations is a,b,c and then d or dialations, horizontal translation, vertical translation

-ve vertical dialation (a)

-ve vertical dialation reflects graph over x-axis

-ve horizontal dialation (b)

-ve horizontal dialation reflects graph over y-axis

what happens if function is written is written without the factorisation (bx-c)

convert into b(x-c/b)

what happen to the operations for the b and c values in the function

operations are reversed e.g. ‘-’ is shift right and x means ‘÷’