2.5-2.6+model+direct+variation-scatter+plots+and+best+fitting+lines

=2.5-2.6= Direct varation

The equation //y// = //ax// repsents **direct varation** between //x// and //y//, and //y// is said to vary directly
with //x//. the nonzero consatnt a is called the **constant of variation.**

the graph of a direct varation equation y = ax is a line with slope a and y-intercept 0. the family of direct varation graphs consists of lines through the orgin such the graph shown below:



= = =**Use ratios to identify driect varation**=


 * Sharks great white sharks have triangular teeth the table below**
 * gives the length of a side of a tooth and the length of the body**
 * for each of six great white sharks.**




 * **tooth length //t (cm)//** || 1.8 || 2.4 || 2.9 || 3.6 || 4.7 || 5.8 ||
 * //**Body length b (cm)**// || 215 || 290 || 350 || 430 || 565 || 695 ||

__EXAMPLE 1:__ > Use the given values of //X// and Y to find the constant or varation.
 * **y**= a**x write driect varation equation**
 * **2** = a(**1**) subsittute **3** for **y** and **//1//** for **//x//**
 * 2= a solve for **a**

=__ 2.6 ____ Draw Scatter Plots and Best-Fitting Lines __= A scatter plot is a graph of a set of data pairs (x,y). There are 3 different types of correlations in scatter plots.
 * The first is a __positive correlation__. Positive correlations occur if //y// tend to increase as //x// increases. Below is an example of a positive correlation. As you can see //x// and //y// are both increasing together.
 * Next, is a __negative correlation__. Negative correlations occur if //y// tends to decrease as //x// increases. Below is an example of a negative correlation. You can see that //y// is decreasing while //x// is increasing.


 * Finally, is the __approximately no correlation__. This happens when the points show no obvious pattern. The points are just scattered anywhere on the graph.

Also in __scatter plots__ you need to know what a correlation coefficient is. A correlation coefficient, denoted by //r,// is a number from -1 to 1 that measures how well a line fits a set of data pairs (x,y). if //r// is near 1, the point lies close to a line with positive slope. If //r// is near -1, the position the points lie close to a line with negative slope. if //r// is near0, the points do not lie close to any line.

Approximating a best fitting line Step 1. **Draw** a scatter plot of the data. Step 2. **Sketch** the line that appears to follow most closely the trend given by the data points. There should be about as many points above the lines as below it. Step 3. **Choose** two points on the line, and estimate the coordinates of each point. These points do not have to be original data points. Step 4. **Write** an equation of the line that passes through the two points from Step 3. This equation is a model for the data.

__Best-fitting Lines__ The best-fitting line is the line that lies as close as possible to all the data points. You can approimate the best-fitting line by graphing it. Above are 4 different steps to help.