# Plotting Functions Worksheets

Mathematical functions might look like a code or another unintelligible language if you don't have the proper knowledge. A function describes how one quantity, the independent variable, depends on another quantity, the dependent variable, usually in the form of an equation or a graph. For example, the function y = x^{2} defines the equation 2x = y. As a rule, the graph of the function y = x^{2} will be a parabola with the point at which y=0 is the vertex. In mathematics, functions are the most fundamental aspect of Calculus. They, like graphs, are used to describe a relationship between two quantities. A function can also be described as a mapping or transformation. The letters f, h, and g are widely used to represent them. This series of lessons and worksheets will help students learn to interpret the meaning and applications of functions that are plotted or fully graphed.

### Aligned Standard: HSS-ID.B.6b

- Trending Step-by-step Lesson- No, not on Twitter! On the premade graph.
- Guided Lesson - More trends here. We look at all the possible trends to give you more experience.
- Guided Lesson Explanation - The key information is at the top of the first page.
- Practice Worksheet - Make sure to focus on the line of best fit.
- Matching Worksheet - Match the graphs and their trends.

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

### Homework Sheets

Do the scattergrams show us a positive trend, negative trend, or no trend?

- Homework 1 - A scatter plot shows a positive trend if y tends to increase as x increases.
- Homework 2 - This scatter plot shows a negative trend.
- Homework 3 - A rise is positive, but a fall is a negative trend.

### Practice Worksheets

Identify the relationship form in one word.

- Practice 1 - State the relationship that each graph displays as positive, negative, or no trend.
- Practice 2 - We can see a fixed relationship. As x increases, y tends to increase. Then we draw the line that best fits the data and notice that this line has a positive slope.
- Practice 3 - Do you remember your line of best fit?

### Math Skill Quizzes

The quizzes stay on par with what you have seen in all the other materials.

- Quiz 1 - Describe the trend of the graphs.
- Quiz 2 - These are ballpark ideas we are looking for here.
- Quiz 3 - What makes this work so well?

### How to Make Sense of a Function

If we talk about functions in terms of sets, it tells how many elements of a non-empty set belong to another (non-empty). If the two sets are A and B, then a function can be written as: f = {(a,b)| for all a ε A, b ε B}.

An important point to remember here is that each input x has a unique output. The input is the domain, whereas the values representing an output are range or co-domain. There are many ways to represent a function, including a graphical and pictorial representation, set of ordered pairs, and tabular form.

As we have discussed, the function is a relationship that indicates input and output values or, in other words, whether the output is a function of the input or not. We can begin to understand the nature of the function by paying attention to the input and output values. If each input has a distinct output, label it a function; otherwise, don't as an input can't be associated with two outputs. Once you have identified a set of values as a function, the next step is to identify the type of function.

**One-to-One** - A function is said to be one-to-one if there is a distinct element of B for each element of A. In other words, it is a version in which the answer never repeats. This type is also named Injective. Consider an example of f(x) = x^{2}; this function is not a one-to-one because the output will be the same if we input the same number with a different (4 in case of 2 and -2). However, if we change the form to f(x) = x – 2, it becomes a one-to-one because there will be a different output for each value.

**Many to One** - A function is said to be many to one if two or more elements of A result in the same element of B. In others, a function in which multiple inputs can give a single output. An example of this can be f(x) = x^{2}; as a result, it will be 4 if we input 2 and -2.

**On-to** - A function is said to be on-to if each element of B has a corresponding value of A such that no element of B is left unpaired. In other words, the range is the elements of set B. This type is also named Surjective. Consider an example, where A = {1, 2, 3}, B = {4, 5} and f = {(1, 4), (2, 5), (3, 5)}. This is an on-to function as the output is the elements of set B.

**One-to-one and On-to** - A function is said to be one-to-one and on-to if it displays the characteristics of both types. In other words, if every element of A corresponds uniquely to B and the output of the function is also B. Another name for this type is Bijective. Consider an example, where A = {1, 2, 3}, B = {4, 5, 6} and f = {(1, 4), (2, 5), (3, 6)}. It follows both characteristics. There is a unique input, and the output is also the element of set B.

### How Do You Analyze Plotted Functions?

Plotting functions is not as technical or difficult as the students believe it to be. However, the real challenge lies when it comes to analyzing plotted functions. A lot of students find it difficult to reach to conclusions and identify the trends in it.

These three simple steps will help you analyze a plot and make deductions about it.

**Step 1: Comprehending the Data** - The first step is to comprehend the data by gaining a basic understanding of the data.

**Step 2: Interpretation of the Data** - While interpreting the data we have to identify the obstacles, examine the climax as well as the resolution of the plot. It is the part where we look at different parts of the story.

**Step 3: Conclusions** - The final step is to conclude and identify the trends of data by looking at the plot.