Comprehensive Guide to Transforming Mathematical Functions

Posted on Aug 22, 2024 in Mathematics

Transformations of Quadratic Functions

The general form of a quadratic function is:

g(x) = af(b(x + c)) + d

This formula describes how to transform the graph of the basic quadratic function y = x². Let’s break down each parameter:

Vertical Transformations

  • a > 1: Vertical Stretch: The graph is stretched vertically by a factor of a. For example, if a = 2, the graph is twice as tall.
  • 0 < a < 1: Vertical Compression: The graph is compressed vertically by a factor of a. For example, if a = 0.5, the graph is half as tall.
  • a < 0: Vertical Reflection: The graph is reflected across the x-axis. If a is negative, the parabola opens downwards.
  • d > 0: Vertical Shift Up: The graph is shifted upwards by d units.
  • d < 0: Vertical Shift Down: The graph is shifted downwards by d units.

Horizontal Transformations

  • b > 1: Horizontal Compression: The graph is compressed horizontally by a factor of 1/b. For example, if b = 2, the graph is half as wide.
  • 0 < b < 1: Horizontal Stretch: The graph is stretched horizontally by a factor of 1/b. For example, if b = 0.5, the graph is twice as wide.
  • b < 0: Horizontal Reflection: The graph is reflected across the y-axis.
  • c > 0: Horizontal Shift Left: The graph is shifted to the left by c units.
  • c < 0: Horizontal Shift Right: The graph is shifted to the right by c units.

Examples of Quadratic Function Transformations

  • g(x) = f(x - 3) - 1: Horizontal shift right 3 units & vertical shift down 1 unit.
  • 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 units & shift up 1 unit.
  • k(x) = -0.5f(-2(x + 3)) + 1: Vertical reflection, vertical compression by a factor of 0.5, horizontal reflection with compression by a factor of 1/2, shift left 3 units & shift up 1 unit.

Linear Inequalities

Consider a scenario where you’re creating 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

You can use Desmos or a similar graphing calculator to visualize the solution set. Remember to use x and y variables for graphing.

Equation

Matrices

To find the reduced row echelon form (RREF) of a matrix using a calculator:

  1. Enter the matrix dimensions.
  2. Enter the matrix values.
  3. Go to the matrix menu and select the matrix you entered (e.g., matrix A).
  4. Go to the math menu within the matrix section and choose”rre”.
  5. Apply the rref function to your matrix (e.g., rref(A)).

Equation

For 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

Solving Systems of Equations Using Matrices

You can represent a system of equations like this:

3x - 7y = 3 4x + 5y = 47

…as an augmented matrix:

3 -7 3 => 1 0 8 4 5 47 => 0 1 3

After performing row operations to reach the RREF, you get the solution: x = 8 and y = 3.

Rational Functions

A rational function can have the following characteristics:

  • Horizontal Asymptote: A horizontal line the graph approaches as x approaches positive or negative infinity.
  • Horizontal Intercept(s): Point(s) where the graph crosses the x-axis (where y = 0).
  • Vertical Asymptote(s): Vertical line(s) where the function approaches positive or negative infinity as x approaches a certain value.

For example, a rational function with a horizontal asymptote at y = -2, horizontal intercepts at (3, 0) and (5, 0), and vertical asymptotes at x = 1 and x = 6 might look like this:

Equation

To graph rational functions, use a graphing calculator like Desmos and adjust the x and y windows for a clear view.

Equation

Cubic Functions

Cubic functions are polynomials of degree 3. They have the general form:

f(x) = ax³ + bx² + cx + d

Key features of cubic functions include:

  • Concavity: Cubic functions can concave up and down, with one inflection point where the concavity changes.

Equation

Quartic Functions

Quartic functions are polynomials of degree 4. They have the general form:

f(x) = ax⁴ + bx³ + cx² + dx + e

Key features of quartic functions include:

  • Concavity: Quartic functions can concave up only or down only, depending on the leading coefficient.

Equation

Power Functions

Power functions have the general form:

y = ax^b

Where a and b are constants.

Direct Variation

In direct variation, as x increases, y increases proportionally.

Inverse Variation

In inverse variation, as x increases, y decreases proportionally.

Growth Factor

The growth factor represents the factor by which a quantity increases over time. To calculate the growth factor:

Growth Factor = Future Value / Current Value

Logarithms

Logarithms are the inverse operation of exponentiation. They are useful for solving equations where the variable is in the exponent.

Example: Finding When Two Populations Will Be Equal

Suppose the population of Africa is 2.854 billion with a growth rate of -1.06% per year, and the population of Europe is 2.436 billion with a growth rate of 1.98% per year. To find out when the populations will be equal, we can set up an equation:

2.854(1 - 0.0106)^t = 2.436(1 + 0.0198)^t

Solving for t involves using logarithms:

  1. Divide both sides by 2.436: 1.172(0.9894)^t = (1.0198)^t
  2. Divide both sides by (0.9894)^t: 1.172 = (1.0198 / 0.9894)^t
  3. Simplify: 1.172 = 1.031^t
  4. Take the logarithm of both sides: log(1.172) = log(1.031^t)
  5. Use the logarithm power rule: log(1.172) = t * log(1.031)
  6. Solve for t: t = log(1.172) / log(1.031) ≈ 5.2 years

Therefore, the populations are projected to be equal in approximately 5.2 years.

Exponential Model

An example of an exponential model is:

406,500(1.0271)^t

Where:

  • 406,500 is the initial value
  • 1.0271 is the growth factor (1 + growth rate)
  • t is the time in years

Percentage Change

To calculate the percentage change, you can use the following formula:

Percentage Change = ((New Value - Old Value) / Old Value) * 100%

Present Value

The present value formula is used to calculate the current value of a future sum of money, given a specific interest rate and time period. The formula is:

PV = FV / (1 + r/n)^(nt)

Where:

  • PV = Present Value
  • FV = Future Value
  • r = Annual Interest Rate (as a decimal)
  • n = Number of times interest is compounded per year
  • t = Time in years

Piecewise Functions

Piecewise functions are defined by different rules over different parts of their domains. For example:

p = f(h) = { $4.00 hourly, 0 < h ≤ 2.75 { $11.00 maximum, h > 2.75

This function describes an hourly wage of $4.00 for up to 2.75 hours, with a maximum earning of $11.00 for any time worked beyond 2.75 hours.

Equation

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