1. Objectives  Graph the relationship between Independent and Dependent Variables.  Interpret Graphs.  Recognize common relationships in graphs.

2. INTRO A well designed graph can convey information quickly and simply. In this section you will develop graphing techniques that will enable you to display, analyze, and model data.

3. IDENTIFYING VARIABLES When you perform an experiment, it is important to change only one variable at a time. Variable - any factor that might affect the behavior of an experimental setup. Independent Variable - is the factor that is changed or manipulated during the experiment. Dependent Variable - is the factor that depends on the independent variable. One way to analyze data is to use a line graph (see Fig. 1-15). This shows how the dependent variable changes with the independent variable.

4. IDENTIFYING VARIABLES Line of Best Fit – a line that best passes through or near graphed data. It is used to describe data and predict where new data will appear on the graph. Problem Solving Strategy p. 16 The Independent Variable is plotted on the Horizontal (x) axis and The Dependent Variable is plotted on the Vertical (y) axis.

5. LINEAR RELATIONSHIPS The line of best fit may be called the Curve of Best Fit for non-linear graphs. Linear Relationship - When the line of best fit is a straight line, as in the figure 1-16, the dependent variable varies linearly with the independent variable. The Linear Relationship can be written as an equation as y = mx + b Slope - the ratio of the vertical change to the horizontal change. To find the slope, select two points, A and B, far apart on the line. The vertical change, or Rise, Δy, is the difference between the vertical values of A and B. The horizontal change, or Run, Δx, is the difference between the horizontal values of A and B.

6. LINEAR RELATIONSHIPS Slope = m = y 2 – y 1 = Rise = Δy x 2 – x 1 Run Δx If y gets smaller as x gets larger, then Δy/Δx is negative, and the line slopes downward. If y gets larger as x gets larger then slopes upward (positive). The y-intercept, b, is the point at which the line crosses the y-axis, and it is the y-value when the value of x is zero.

7. NONLINEAR RELATIONSHIPS When the graph is not a straight line, it means that the relationship between the dependent variable and the independent variable is not linear. There are many types of nonlinear relationships in science. Two of the most common are the quadratic and inverse relationships. Quadratic Relationship - exists when one variable depends on the square of another. The resulting graph is a Parabola. It can be represented by the following equation y = ax 2 + bx + c Inverse Relationship – exists when a variable depends on the inverse of another variable. The resulting graph is a Hyperbola. It can be represented by the following equation y = a / x

8. NONLINEAR RELATIONSHIPS There are various mathematical models available apart from the three relationships you have learned. Examples include: sinusoids—used to model cyclical phenomena; exponential growth and decay—used to study radioactivity. Combinations of different mathematical models represent even more complex phenomena.

9. PREDICTING VALUES Relations, either learned as formulas or developed from graphs, can be used to predict values you have not measured directly. It is important to decide how far you can extrapolate (to estimate to values outside the known range) from the data you have. Physicists use models to accurately predict how systems will behave: what circumstances might lead to a solar flare, how changes to a circuit will change the performance of a device, or how electromagnetic fields will affect a medical instrument.

10. QUESTION 1 Which type of relationship is shown in the following graph? Inverse

11. QUESTION 2 What is a line of best fit? The line drawn closer to all data points as possible, is called a line of best fit. The line of best fit is a better model for predictions than any one or two points that help to determine the line.

12. QUESTION 3 Which relationship can be written as y = mx? Linear relationship is written as y = mx + b, where b is the y intercept. If y-intercept is zero, the above equation can be rewritten as y = mx