Comprehensive Guide to Transforming Functions: Quadratic, Rational, Cubic, and More
Transformations of Functions
Quadratic Function:
The graph of the basic quadratic function Y=x^2 can be transformed using the following parameters:
g(x) = af(b(x+c)) + d
- a:
- a > 1: Vertical stretch by a factor of ‘a’.
- 0 < a < 1: Vertical compression by a factor of ‘a’.
- a = -1: Vertical reflection across the x-axis.
- b:
- b > 1: Horizontal compression by a factor of 1/b.
- 0 < b < 1: Horizontal stretch by a factor of 1/b.
- b = -1: Horizontal reflection across the y-axis.
- c:
- c > 0: Horizontal shift left by ‘c’ units.
- c < 0: Horizontal shift right by ‘c’ units.
- d:
- d > 0: Vertical shift up by ‘d’ units.
- d < 0: Vertical shift down by ‘d’ units.
Example: Describe how the graph of f(x) = 2(x-3)^2 + 8 compares to the graph of Y=x^2.
Answer: The graph of f(x) is a vertical stretch by a factor of 2, a shift right by 3 units, and a shift up by 8 units compared to the graph of Y=x^2.
Linear Inequality:
Example: Create a snack mix using trail mix and blueberries. The mix can contain at most 700 calories, at least 7 grams of fiber, and 12% calcium.
Let:
- t = amount of trail mix
- b = amount of blueberries
Use Desmos to graph the inequalities (using x and y variables instead of t and b).
Matrix:
To solve a system of equations using matrices:
- Enter the matrix: Press 2nd, MATRIX, EDIT, enter the dimensions, and enter the values.
- Find the reduced row echelon form (RREF): Press 2nd, MODE, 2nd, MATRIX, MATH, choose RREF, 2nd, MATRIX, choose the matrix name (e.g., A).
Example: Solve the system of equations:
3x – 7y = 3
4x + 5y = 47
Matrix Form:
[3 -7 | 3]
[4 5 | 47]
RREF:
[1 0 | 8]
[0 1 | 3]
Solution: x = 8, y = 3
Rational Function:
A rational function can have the following characteristics:
- Horizontal asymptote: A horizontal line that the graph approaches as x approaches positive or negative infinity.
- Horizontal intercepts (x-intercepts): Points where the graph crosses the x-axis.
- Vertical asymptotes: Vertical lines where the function approaches positive or negative infinity.
Example: Write the equation of a rational function that has a horizontal asymptote at y = -2, horizontal intercepts at (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))
Cubic Function:
A cubic function is a polynomial function of degree 3. Its graph can have up to two concavity changes and one inflection point.
Quartic Function:
A quartic function is a polynomial function of degree 4. Its graph can have up to three concavity changes.
Power Function:
A power function has the form Y = ax^b, where ‘a’ and ‘b’ are constants.
- Direct variation: As x increases, y increases.
- Inverse variation: As x increases, y decreases.
Exponential Model:
An exponential model represents growth or decay over time. The general form is A = P(1 + r/n)^nt, where:
- A = future value
- P = present value
- r = annual interest rate (as a decimal)
- n = number of times interest is compounded per year
- t = time in years
Logarithms:
Logarithms are used to solve for the exponent in an exponential equation. The equation logba = c is equivalent to b^c = a.
Growth Factor:
The growth factor is the factor by which a quantity increases over a specific period. It is calculated by dividing the future amount by the current amount.
Piecewise Function:
A piecewise function is a function defined by multiple sub-functions, each applying to a different interval of the input domain.
Example:
p = f(h) =
- 2h, if 0 <= h <= 40
- 2.5h, if h > 40
