Interview Flashcards
(13 cards)
- How do you sketch the graph of y= af/k (x- d)] +c when you have the graph of y= f(x).
- (x-d) = horizontal shift (d>0 shift right, d<0 shift left)
k = horizontal stretch/compression
- k>1 graph compressed [by 1/k]
- (-1)<k<1 stretched
- k<0 reflection on y-axis
a = vertical stretch/compression
- a>1 stretched taller
- 0<a<1 compression shorter
- a<0 reflection on x-axis
c = vertical shift
- c>0 upwards translation
- c<0 downwards translation
- How to determine whether two polynomials are equivalent?
- Expand and simplify each polynomial to standard form
- if both polynomials are same = equivalent (same coefficients of corresponding powers - Substitute value for x (test point)
- if outputs are same for every value of x inputted, polynomials are likely equivalent (3 tests) - Factor to see if they factor into the same expression
- GCF, simple quadratic (a=1), general quadratic, group terms (+)(+), difference of square (a^2-b^2)
- What is a rational function and how will you determine its simplified form? How can we add, subtract, multiply, and divide rational expressions?
f(x) = 1/x. x≠ o
To simplify
1. factor numerator & denominator (GCF)
2. cancel common factors in numerator & denominator
3. state restrictions (anything in denominator)
Adding and subtracting
1. find common denominator (LCD), multiply the numerator and denominator
2. combine numerators
3. simplify by canceling common factors
Multiply and divide
1. Factor numerator and denominator
2. multiply numerator and denominators together and cancel out common factors
* To divide, find reciprocal first)
- What are the steps to find the maximum or minimum value of a quadratic function?
(When the equation is in standard form and when the equation is in factored form)
Completing the square (f(x)= ax^2+bx+c)
1. factor from first two terms
2. take coefficient of x (b), divide it by 2, then square
3. add & subtract value inside parentheses
4. function will be in vertex form
- value inside parentheses (d) = x
- (c) = y
- a determines if it opens up or down
Factored Form
1. find axis of symmetry (x= r1+r2/2) [roots]
2. substitute x value obtained to find y
3. write coordinates and whether max or min
*perf square trinomial
- Explain how to find the point of intersection of a linear function and a quadratic function graphically and algebraically? Do you think that a linear function always intersects a quadratic function in two places? Why or why not?
Graphically
1. plot both functions (y= mx+b & y=ax^2+bx+c)
2. find intersection points (f(x)=g(x))
Algebraically
1. set the equations to equal each other
2. move all terms to one side (everything=0) and solve for x
-(factor until u get 2 roots), complete square, vertex
3. substitute x back into equation to find y
Liner functions don’t always intersect at two places
- if d = b^2-4ac is positive, two real solutions = two intersections
- if d = 0, one real solution = one intersection
- if d = negative, no real solutions = no intersections
- How to determine the solutions to a quadratic equation? How to use the discriminant to determine the number of solutions of a quadratic equation?
Quadratic formula [x=-b±√(b²-4ac))/(2a)]
if D>0, two distinct real solutions
-parabola crosses x-axis at two points
if D=0, one real solution
-tangent to x-axis (bounces off x-axis)
if D<0, no real solutions
-lies above or below x-axis
- How to determine the trigonometric ratios for any angle 0, where 0° ≤ 0 ≤ 360°?
Unit circle (radius of 1)
- x=cosθ , y=sinθ , tanθ= sinθ/cosθ
CAST
- Q1: all Q2: sin+, cos- , tan- Q3: tan+ , sin- , cos- Q4: cos+ , sin- , tan-
Trig ratios for standard angles
-sin45= 1/√2, cos45= √2/2, tan45= 1
-sin30= 1/2, cos30= √3/2, tan30= 1/√3
-sin60= √3/2, cos60= 1/2, tan60= √3/1
Find reference angle (smallest angle with x-axis) (subtract or add)
-use quadrant to determine sign
- Describe how sine, cosine and tangent functions can be verified using the CAST rule.
Determine quadrant where θ lies
Apply CAST rule to determine signs of functions
Use reference angles to find exact value
-C: (360-θ), S: (180-θ), T: (180+θ)
- How to decide when to use the sine law or the cosine law to solve a problem?
Sine law
-when given 2 angles and one side (AAS & ASA)
-when given 2 sides and a non-included angle (SSA)
- a/sinA = b/sinB = c/sinC
Cosine law
-given 2 sides and included angle (SAS)
-given 3 sides (SSS)
-c^2 = a^2 + b^2 - 2ab * cosC
- Describe the characteristics of exponential functions and their graphs.
Domain
-all real numbers (+∞,-∞), no restrictions for x
Range
- a>0, range= (0, +∞), positive values only
-a<0, range= (-∞,0), negative values only (never touches x-axis)
Y-intercept
-at (0,a) b/c y= a* b^0 =a
Horizontal asymptote
- @ y=0, as x -> +∞ or x -> -∞, function approaches x-axis
Growth or decay
-exponential growth: b>1, increases rapidly as x increases, rises from left to right
-exponential decay: 0<b<1, decreases rapidly as x increases, falls from left ot right
Behaviour
-b>1, x -> +∞, y -> ∞ (rapid increase)
x -> -∞ or y ->0 (approaching x-axis)
-0<b<1, x -> +∞, y -> 0 (approaching x-axis)
x -> -∞, y -> ∞ (rapid decrease)
No symmetry
- Compare exponential functions with linear and Quadratic functions.
Rate of change
-grow/decay faster than linear & quadratic functions, shows multiplicative change
-change is constant
-grows at variable rate that accelerates as you move away from vertex
Graph shape
-curved shape, increasing or decreasing rapidly, has horizontal asymptote
-straight line w/ constant slope, no curve or asymptotes
-parabola, symmetrical around the vertex
End behaviour
-grow or decay infinitely fast as x -> +∞ or x -> -∞
-constant slopes that extend infinitely both directions
-one side heading towards to +∞ and the other -∞ (depending a)
- How can exponential functions model growth and decay?
Used to model situations where changes happen at a rate proportional to the current value
Exponential growth
-when a quantity increases rapidly overtime w/ each step building from the last one
ex. population growth, population increases by fixed percentage each year
Exponential decay
-quantity decreases at a rate proportional to its current value
ex. value of a car decreasing each year
Accurately represents processes where changes accelerate or decelerate over time
- What are the steps to determine the equation of a sinusoidal function from its graph?
y=2sin(3(x-1))+4
Find amplitude (a= maxy-miny/2)
-midline to the max/min point, halfway between max & min points
Determine vertical shift (EOA= maxy+miny/2)
-value of midline (horizontal)
Find period & horizontal stretch/compression
-period: one peak to another, p= 360/b
b: 360/p
Identify horizontal shift
-if graph starts from max, shift cosine graph 0 b/c it starts at max
Write equation
If the graph starts at zero or the midline, the sine function might be more appropriate
