# Linear Quadratic and Exponential Models Worksheets

When we are modelling data to help us make concrete and solid decisions, we will common across three different types of models. We can identify the type of data that we are working with, it helps us better understand how to analyze this data. You can identify the model that is being presented from the composition of the equation. You will look for the visibility of exponents and note the powers to which they are being presented. You can also tell the nature of these models based on how it bends or does not bend on a coordinate graph. These worksheets and lessons teach students how to create models of data in various formats.

### Aligned Standard: HSF-LE.A.2

- Function It All Up Step-by-step Lesson - You are given a table and asked to write a linear, quadratic, or exponential function for it.
- Guided Lesson - We build on the lesson and now we introduce some negative numbers too.
- Guided Lesson Explanation - The second one drags out a bit, but it makes sense in the end.
- Practice Worksheet - This page took two hours for one students of a teacher that wrote in. You might want to do this over several days.
- Matching Worksheet - I give you one of each form to find. You can reverse engineer it, when using this type of worksheet.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Modeling data is not always a simple task. Think before attempting these questions.

- Homework 1 - The pattern of the y-values lets us know if the function is linear, quadratic, or exponential.
- Homework 2 - This allows us to compare each y value directly.
- Homework 3 - If the first difference between successive y-values is equal, this will be a linear function. If the second differences between the successive y-values are equal, the function is quadratic .

### Practice Worksheets

Model all of the data in three different formats.

- Practice 1 - For each problem, write a:
a) linear (y = mx + b),
b) quadratic (y = ax
^{2}+ bx + c) or c) exponential (y = a(b)^{x}) function that models the data. - Practice 2 - Look at this table and write a linear (y = mx + b),
Quadratic (y = ax
^{2}bx + c), or exponential (y = a(b)^{x}) function that models the data. - Practice 3 - Since the x-values are consecutive and straight forward, it makes it much easier.

### Math Skill Quizzes

Yet more modeling for you. This is a very lucrative skill to have in the real world.

- Quiz 1 - You can test for exponential functions by finding the ratios between successive y-values. If the ratios are equal, it is exponential.
- Quiz 2 - Find the first and second differences in the table.
- Quiz 3 - If the second differences are not all equal, the function can't be quadratic.

### Understanding Linear, Exponential, and Quadratic Models of Data

Linear, quadratic, and exponential functions have independent value in the field of mathematics. Each of these models have different equations, let us discuss each of them differently.

**Linear Model** - The linear model should have an equation with the format y = mx + b, where m is the slope, while b is the y-intercept. You will notice that this model lacks an exponent on the x exponent. They make a straight line when graphed. When the independent variable changes by a constant amount, they have constant first differences.

**Exponential Model** - These types of models should have an equation with the format of y = a^{x}. Please note that an exponent is always present with these models, most commonly the variable (x) serves as the base for it. The graph goes steeper with an increase in value. In general, the graphs begin with subtle changes and then grow or decay explosively. When the independent variable changes by a constant amount, they have constant ratio.

**Quadratic Model** - The quadratic model should have an equation with the format y = [ax]^{2}. They have an exponent raised to the highest power. They make a parabola if graphed. When the independent variable changes by a constant amount, they have constant second differences.

As we have discussed, there are two distinct options you have when classifying these types of models. You can look at the nature of the equation and see if there are exponents presents and if so, is anything raised to the highest power. You can also look at what they look like on a graph. A straight line, a drastic curve, or a parabola are dead giveaways.