Functions (topic 2) Flashcards
Functions
One variable depends on the other
Domain
Is the set of all valid input values (x-values) for a function. For this course, domain is limited by the following illegal operations:
- division by zero
- roots of negatives (greater than or equal to zero)
- logs of zero and negatives (must be positive)
- Certain values of tan
Range
is the corresponding set of output values (y-values) for function. The graph and / or knowledge of asymptotes, maxima, minima should help determine the range.
Composite functions
(fog)(x) is a composite function. It means f following g or f(g(x)). Start evaluating from the inside. In general, order matters in a composite function.
Inverse functions
inverse f(x) is NOT 1 / f(x)
Slope and y-intercept form
y = mx + c
Point-slope form
y - y1 = m(x - x1)
Slope or gradient
m = Δy / Δx = (y2 - y1) / (x2 - x1)
To find y-intercept
Sub 0 for x
To find the x-intercept
Sub 0 for y
Quadratic function
Highest power x^2
Types of quadratic functions
- Vertex form
- Factored form
- Trinomial / expanded form
Vertex form
y = a(x - h)^2 + k
- the vertex is (h, k)
- the a-value determines the direction of opening and steepness
Factored form
y = a(x - p)(x - q)
- the values of p and q are the x-intercepts
- the vertex and axis of symmetry lie halfway between the x-intercepts
- the a-value determines the direction of opening and steepness
- parabolas that do not have x-intercepts cannot be written in this form
Trinomial / expanded form
y = ax^2 + bx + c
- c is the y-intercept
- the a-value determines the direction of opening and steepness
- the axis of symmetry is x = -b / (2a) (x-coordinate of vertex)
- can be converted to “vertex form” through completing the square, using the axis of symmetry, or using differentiation techniques
The discriminant
Δ = b^2 - 4ac
In a quadratic equation, discriminant tells us how many real solutions exist. In a quadratic function, discriminant tells us how many x-intercepts exist.
If Δ < 0
There are no x-intercepts or real solutions
If Δ = 0
There is one (repeated) x-intercept or real solution
If Δ > 0
There are two distinct x-intercepts or real solutions
Finding the horizontal asymptote
imagine y-value as x gets large (think lim x->infinity)
Finding the vertical asymptote
what x-value makes the bottom zero (domain restriction?)
Function transformations
- f (x - c): translate (shift) right c units
- f (x) + d: translate (shift) up d units
- pf(x): stretch vertically by a scale factor of |p|
(note: negative p values cause a reflection through the x-axis) - f (qx): stretch horizontally by a scale factor of |1 / q|
(note: negative q values cause a reflection through the y-axis)
