Knowt
IB Functions
Linear Functions
Forms
y = mx + c
“Gradient-intercept form”/”Slope-intercept form”
m is the gradient/slope
c is the y-intercept
Example:
y = 2x - 1
Gradient = 2
y - intercept at y = -1
ax + by = c
“Standard Form”
Example:
y = (2/3)x - 1
-(2/3)x + y + 1 = 0
Answer: -2x + 3y + 3 = 0
y - y1 = m(x - x1)
“Point - slope form”
useful for finding an equation from slope (m) and point (x1,y1)
Example:
Find line with slope 2 that passes through (1,3)
y - 3 = 2(x - 1)
y - 3 = 2x - 2
Answer: y = 2x + 1
Relationships
Parallel lines: two or more lines that lie in the same plane and do not intersect. Have same slope
Perpendicular lines: lines that intersect at a 90-degree angle
Intersection: found by “setting equal to each other”; substitution
Example:
Intersection of y = 2x -1 & y =-x + 8
2x - 1= -x + 8
3x = 9
x = 3
Equation 1: y = 2(3) - 1
y = 5
Answer: (3,5)
Function Concepts
Function: series of operations that will output one specific value (y) for any input (x)
Notation: y = 4x - 3 or f(x) = 4x - 3
Domain: set of x values that can be plugged in. can be limited by division by zero, log of 0 or negative, sqrt of negative
Example:
Domain of f(x) = ln(x + 1)
x + 1 > 0
x > -1 or {xR|x > -1} or (-1,∞)
A restricted domain can be used to make a one-to-many “function” into a one-to-one function
Example:
y = sqrt(x) is one-to-many
f(x) = sqrt(x); x0 is a function
Range: set of y values that can be outputted
Example:
Range of g(x) = 2x2 - 3
x2 > = 0
g(x) > = -3 or [-3,∞)
Inverse: reverses the effect of the original function; x and y are switched only a function if original function is one-on-one, if output of original function is plugged in, the input will be returned
graph is reflected over y = x
Notation: f-1(x)
Example:
f(x) = 2x - 4
y = 2x - 4
x = 2y - 4
x + 4 = 2y
y = (1/2)x + 2
f-1(x) = (1/2)x + 2
Self-inverse: A function that is an inverse of itself. If f(x) = f-1(x) or f(f(x)) = x
Example:
f(x) = 0.5/x is a self - inverse function
f(f(x)) = 0.5/f(x) = 0.5/(0.5/x) = 0.5*x/0.5 = x
Even function: f(-x) = f(x)
Odd function: f(-x) = -f(x)
Graphs
Features
Absolute Maximum/Minimum: The highest/lowest point on a graph
Relative Maximum/Minimum: “turning points” of the graph
Intercepts: where the graph crosses the x - axis (y = 0) or the y - axis (x = 0)
Line of symmetry: mainly quadratics, splits the graph in half
Vertex: Max or min points for quadratic function
Zero/Roots: Where y = 0, in other words, the x - intercept
Asymptotes: A line that a graph will get infinitely close to but will never touch, as x or y tends to infinity
Vertical Asymptotes: x value of a restricted domain E.X. divide by 0
Horizontal Asymptotes: Find what the graph get closer to by plugging in very large/very small values for x
Composite Functions
Composite Function: plug in a whole other function as x into a function, creating a new function of x
Notation: (fg) (x) = f(g(x))
Example:
f(x) = 6x - 5 & g(x) = x2 - x ,find (gf)(1)
f(1) = 6(1) - 5 = 1
g(f(1)) = g(1) = (1)2 - 1 = 0
Composite functions can be used to check inverse functions. Inverse functions must follow:
f(f-1(x)) = x
Quadratics
Quadratics: polynomial with degree of 2
Form
f(x) = ax2 + bx + c
“Standard Form”
y-intercept: (0, c)
c is known as the constant
Axis of symmetry: x = -b/2a
f(x) = a(x-p)(x-q)
“Factored Form”
Obtained through factoring
Roots: x = p & x = q
f(x)=a(x-h)2+k
“Vertex Form”
Obtained through completing square
Vertex: (h, k)
a: leading coefficient
Factorization
Factorization: converting from standard form to factored form
When a = 1:
(x+p)(x+q) = x2 + (p+q)x + pq
Find numbers p and q that add up to b and multiply to c
Example:
x2 + 3x + 2
p + q = 3
pq = 2
p = 1, q = 2
(x+p)(x+q) = (x+1)(x+2)
When a1:
(mx+p)(nx+q) = ax2 + bx + c
General idea: We must find m, n, p, c such that mn=a, pq=c, mq+np=b. We can do this with various methods (e.g. Star Method, Funnel Method) or trial-and-error.
Factoring by grouping: One method of factoring involving separating quadratic into groups and finding GCF
Example:
2x2+11x+12
= (2x2 + 3x) + (8x + 12) (Grouping)
= x(2x + 3) + 4(2x + 3) (Factor out GCF)
= (x + 4)(2x + 3) (Combine like terms)
Solving
Zero product property: solve either bracket equal to zero to find 2 roots
Example:
(x + 4)(2x + 3)
x + 4 = 0
x = -4
2x + 3 = 0
2x = -3
x = -3/2
Answer: x = -4 or x = -3/2
Completing the square: converting from standard form to vertex form
ax2+bx+c=a(x-h)2+k
Example:
4x2 + 20x - 24 = 0
x2 + 5x - 6 = 0 (divide by a)
[x2 + 5x + (5/2)2] - 6 = (5/2)2 (add [b/(2a)])
[x + (5/2)]2 = (5/2)2 + (24/4) (rewrite as squared term)
[x + (5/2)]2 = 49/4
x + (5/2) = (7/2) (take the square root, positive and negative roots!)
x = -(5/2) (7/2)
x = 1 or x = -6
Generalized to derive the quadratic formula!
Discriminant: from the quadratic formula. determines how many roots a quadratic has
∆ = b2 - 4ac
If ∆ > 0, there are two distinct roots
If ∆ = 0, there is one repeated root
If ∆ < 0, there are no real roots
Hidden Quadratics
u - substitution: replacing u with a function to make a solvable quadratic
Example:
2e2x + 5ex = 3
2(ex)2 + 5(ex) - 3 = 0
u = ex
2u2 + 5u - 3 = 0
(2u - 1)(u + 3) = 0
2u - 1 = 0
2u = 1
u = 0.5
ex = 0.5
x = ln(0.5)
u + 3 = 0
u = -3
ex = -3 (undefined)
x = ln(0.5)
Rational functions
Rational functions: a fraction where both the numerator and the denominator are polynomials
defined by the location of its asymptotes:
f(x) = (ax+b) / (cx+d)
Vertical Asymptotes: finding an x value that cannot be used, i.e. that makes the denominator 0
cx + d = 0; x = -d/c
Horizontal Asymptotes: finding a value that the fraction never reaches, but gets infinitely close to. Plugging in a very large x value shows that the b and d become negligible. f(x) tends to ax/cx or just y = a/c
f(x) = (ax2 + bx + c) / (dx + e): Degree of numerator > degree of denominator, can result in oblique asymptote
Oblique (diagonal) asymptote: Divide using long division
f(x) = (ax + b) / (cx2 + dx + e): Denominator rises much faster, so horizontal asymptote is 0
Reciprocal functions: f(x) = 1/x
horizontal and vertical asymptote at 0
self-inverse, symmetrical about y = x, f-1(x) = f(x)
Exponential and Logarithmic Functions
Exponential Function: variable is the exponent
f(x) = a(b)x + c
b - base
Logarithmic Function: Inverse of exponential function
f(x) = a*logbx + c
b - base
Example:
f(x) = ex
f-1(x) = ln(x)
Transformations
Translations
y = f(x) + b: adding b to every y-coordinate, shifting up by “b” units
y = f(x) - b: subtracting b from every y-coordinate, shifting down by “b” units
y = f(x - a): taking y-coordinate from the original x that is a to the right, shift right by “a” units
y = f(x + a): taking y-coordinate from the original x that is a to the left, shift left by “a” units
Stretches
y = p * f(x): multiply every y-coordinate by p, stretching vertically by scale factor “p”
y = f(qx): taking y-coordinate from the value qx, stretching horizontally by scale factor “1/q”
Reflections
y = -f(x): change the sign of every y-coordinate, reflect over the x-axis
y = f(-x): take the y-coordinate from the -x value, reflect over the y-axis
Horizontal translations and stretches are counterintuitive
Absolute Value
y = |f(x)|: Replace any section of the graph that’s below the x-axis with a reflection across the x-axis
y = f(|x|): The portion of the graph to the right of the y-axis is reflected over the y-axis, replacing what was on the left
Polynomials
Polynomials: general term for functions in the form f(x) = anxn + an-1xn-1 +... + a0
Degree: the largest exponent of a polynomial; n
Examples: cubics - 3, quadratics - 2, quartics - 4
Remainder Theorem: If a polynomial, P(x), is divided by (x-c), the remainder is P(c)
Factor Theorem: (x - c) is a factor of polynomial P(x) if and only if P(c) = 0
Division Methods
Synthetic division: Use this method when dividing by (x - c)
Long division: More general method, a bit more time-consuming
Roots
Sum of roots: -an-1/an
Product of roots: (-1)na0/an
Example:
Quadratic
f(x) = ax2 + bx + c
Sum of roots: -b/a
Product of roots: c/a
A polynomial of degree n has exactly n roots (imaginary and real)
Conjugate roots: If z is a root, then z* is a root (z = a+bi & z* = a - bi)
