Transformations of Functions and Mathematical Concepts
Transformations: g(x) = af(b(x+c))+d
Parameters and Their Effects
a
- a > 1: Vertical stretch
- 0 < a < 1: Vertical compression
- a < 0: Vertical reflection
b
- b > 1: Horizontal compression
- 0 < b < 1: Horizontal stretch
- b < 0: Horizontal reflection
c
- c > 0: Shift left
- c < 0: Shift right
d
- d > 0: Shift up
- d < 0: Shift down
Examples
g(x) = f(x-3) – 1: Horizontal shift right 3 & vertical shift down 1
h(x) = -f(x): Vertical reflection
j(x) = f(-2(x+3)) + 1: Horizontal reflection, horizontal compression by a factor of 1/2, shift left 3 & shift up 1
k(x) = -0.5f(-2(x+3)) + 1: Vertical reflection, vertical compression by 0.5, horizontal reflection with compression by 1/2, shift left 3 & shift up 1
Linear Inequality: Snack Mix Example
Create a snack by mixing trail mix & blueberries. It can contain at most 700 calories, at least 7 grams of fiber, and 12% calcium.
Let t = trail mix and b = blueberries.
Use Desmos to graph (only use x & y variables).
Matrices
Use a calculator: 2nd matrix, edit, enter dimensions, enter values, 2nd mode, 2nd matrix, math (choose rref), 2nd matrix, names, choose A
Example:
2 2 2 0 10 1 0 0 2
2 4 19 35 = 0 1 0 3
10 2 4 30 0 0 1 1
System of Equations
Example:
3x – 7y = 3 3 -7 3 = 1 0 8
4x + 5y = 47 4 5 47 = 0 1 3 (so X = 8, Y = 3)
Use matrices to solve.
Rational Function
A rational function has a horizontal asymptote at y = -2, horizontal intercepts of (3,0) and (5,0), and vertical asymptotes at x = 1 and x = 6.
Equation:
f(x) = -2(x-3)(x-5) / (x-1)(x-6)
Graphing a rational function (use Desmos), be sure to adjust X & Y windows.
Power Function
Direct variation (as X increases, Y increases)
Inverse variation (as X increases, Y decreases)
Cubic Function
Concave up
Concave down
Concaves up & down & 1 inflection point
Percentage Change
Example: 9-year percent change
1123.33(1 + 0.081)^1×9 = 2.1057 – 1 = 1.0157%
Creating Exponential Functions: y = ab^x
An admin estimated the 2015 population to be 406,500 & projected the 2050 population to be 1,035,500.
Frequency (take the total, subtract 1 & move the decimal right 2 spots)
1035500 / 406500 = b^35 (years)
b = 2.5474
(2.5474)^1 divided by 35 = 1.0271
So the model would be P = 406500(1.0271)^t
Forecast Population
Forecast the 2025 population then compare it to the department projection.
P = 406500(1.0271)^10 years from 2015
P = 406500(1.3066) = 531,133
Comparing: 531133 – 527859 = 3274
Means the projection is off by 3274
Logarithms
Future amount: $3000
Current amount: $2200
= 1.3636(1.039)t
log1.3636 / log1.039 = t = 8.1061 round this to 8
8.1061 becomes 0.1061 x 12 (to get the months) = 1.2732 months, round to 1
Then add months to the years: 8 years + 1 month = approximately 8 years and 1 month
Quartic Function
Concavity changes 0 or 2 times
Quadratic Function
Concaves up only or down only
Growth Factor
A = future value, R = percentage rate (move decimal left 2 places)
P = present value, T = time (number of years)
N = frequency
Frequencies are: annual (1), semi-annual (2), quarterly (4), monthly (12), daily (365)
