Functions: Mathematical Models |

**Mathematical Models or Function Templates**

Mathematical functions are grouped into "families" or sets. **Each "family"** of functions has a **unique shape** (graph) and an equation (or function rule of correspondence) of specific form associated with it. For example, any **quadratic function** will, when graphed, show a **parabola**. The equation that describes this parabola will look like this (in **standard form**):

f(x) = **a** (x - **h**)^{ 2} + **k**

and like this (in **general form**):

f(x) = **a**x^{ 2} + **b**x + **c**

An absolute value function when graphed, will show a vee (V) and its equation in standard form, will look like this:

f(x) = **a** | x - **h** | + **k**

and so on.

The **colored** entries in the function rule are called **parameters**. It is their values that situate the unique set of points or locus (path) of the function in question.

**Parameters**

The parameter "*slots*" in a function rule are what we input to move the function around and determine its shape -- how quickly it rises or falls, where the vertex and the asymptotes are located.

A function rule is very much like this form:

f(x) = |
a |
| x - |
h |
| + |
k |

Once we choose what to put in the colored cells or to input for **a, h **and** k** -- we have fully determined the properties of the particular function.

Below,we see images of the "families" of functions we will discuss in the lessons of MathRoom Functions. Each is accompanied by a typical "function rule template" form for the function rule in standard form. The subsequent lessons in this MathRoom will explain the particular properties of the functions discussed.

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**1/ Linear Function **(first degree)

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**2/ Quadratic Function **(second degree)

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**4/ Greatest Integer or Step Function**

**6/ First-Degree Rational Function**

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Notice the similarity between the 2 curves. If we haul the red y = sin x curve back (left) a quarter of a period (o/2 units), it will look exactly like the cosine curve. These two trig functions are related through the right triangle since the acute angles are complementary (add to 90º) -- so, the cosine of one will equal the sine of the other.

**10/ Tangent Function**

- Notice the asymptotes (vertical straight lines).

- They occur because division by zero is undefined and in this case, limits to infinity.

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The lessons in MathRoom Functions will explain the details and properties of each of these families or sets of functions.

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