# Comprehensive Guide to Transforming Mathematical Functions

## 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.

## Matrices

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

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

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:

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

## 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.

## 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.

## 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:

- Divide both sides by 2.436:
`1.172(0.9894)^t = (1.0198)^t`

- Divide both sides by (0.9894)^t:
`1.172 = (1.0198 / 0.9894)^t`

- Simplify:
`1.172 = 1.031^t`

- Take the logarithm of both sides:
`log(1.172) = log(1.031^t)`

- Use the logarithm power rule:
`log(1.172) = t * log(1.031)`

- 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.