GRAPHS AND RELATIONSHIPS   A scientist studies data gathered from an experiment in an attempt to establish a (mathematical) relationship. Common relationships: Linear (or directly proportional), square, inversely proportional, inverse square. The clearest and simplest way to see a relationship is to plot a graph.

Why be concerned about control variables? Relationship Variables: the independent variable - the one you are testing the dependent variable - the one you are measuring the controlled variables - all other variables that are kept constant (unchanged) Why be concerned about control variables?

DRAWING the GRAPH Be sure to: Independent variable (usually plotted on the x axis) the variable that you have control over Dependent variable (usually plotted on the y axis) this is the quantity that you have to measure and is affected by how you alter the independent variable. Label the axes and use the units Sensible scale please Plot the points Draw a best fit line If it looks like it should be a straight line then use a ruler If it looks like a curve then draw a best fit freehand curve

WHAT’S the RELATIONSHIP? If the best fit line is straight (you used a ruler) then the relationship is called ‘linear’ or ‘proportional.’ If the straight best fit line also goes through the origin it can be called ‘directly proportional’ relationship. If the best fit line is curved and not straight, then the relationship is ‘non-linear.’

Example 1: Distance Time graph:

What are the units?

What are the units?

Ask yourself what is the relationship. i. e. is y α x Ask yourself what is the relationship? i.e. is y α x? or do I need to linearize? y is proportional to x

What is the gradient? 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 4 2 =2 Rise: Run:

What are the units of the gradient? 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 4 2 =2 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑦−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 𝑥−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 Rise: 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑚 𝑠 𝑖.𝑒. 2 𝑚 𝑠 or 2 ms-1 Run:

What is the equation of a straight line? The equation of a straight line is: Y = m x + c

Back to the distance time graph: What is the equation of the following? All straight line graphs are of the form y = mx + c 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 4 2 =2 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑦−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 𝑥−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 Rise: 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑚 𝑠 𝑖.𝑒. 2 𝑚 𝑠 or 2 ms-1 Run:

Back to the distance time graph: What is the equation of the following? All straight line graphs are of the form y = mx + c 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑦 𝑤𝑖𝑡ℎ: 𝑦=2𝑥+0 Rise: Run:

Back to the distance time graph: What is the equation of the following? All straight line graphs are of the form y = mx + c 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑦 𝑤𝑖𝑡ℎ: 𝑑=2𝑥+0 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑥 𝑤𝑖𝑡ℎ: Rise: Run:

Back to the distance time graph: What is the equation of the following? All straight line graphs are of the form y = mx + c 𝑑=2𝑡+0 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑥 𝑤𝑖𝑡ℎ: Rise: Run:

Back to the distance time graph: What is the equation of the following? All straight line graphs are of the form y = mx + c 𝑇ℎ𝑖𝑠 𝑏𝑒𝑐𝑜𝑚𝑒𝑠: 𝑑=2𝑡+0 𝑑=2𝑡 Rise: Run:

What equation has both d & t in it? 𝑣= 𝑑 𝑡 𝑑=2𝑡 a= 𝑣 𝑡

Example 2: Velocity Time graph:

What are the units?

What are the units?

Ask yourself what is the relationship. i. e. is y α x Ask yourself what is the relationship? i.e. is y α x? or do I need to linearize? y is proportional to x

What is the gradient? 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= (20−8) (5−2) = 12 3 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 𝑅𝑖𝑠𝑒 𝑅𝑢𝑛 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= (20−8) (5−2) = 12 3 =4 Rise: Run:

What are the units of the gradient? Rise: Run: 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡= (20−8) (5−2) = 12 3 =4 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑦−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 𝑥−𝑎𝑥𝑖𝑠 𝑢𝑛𝑖𝑡 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑢𝑛𝑖𝑡𝑠: 𝑚𝑠 _1 𝑠 𝑖.𝑒. 4 𝑚𝑠−1 𝑠 or 4 ms-2

What is the equation of the following? Rise: Run: All straight line graphs are of the form y = mx + c 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑦 𝑤𝑖𝑡ℎ: 𝑦=4𝑥+0

Back to the distance time graph: What is the equation of the following? Rise: Run: All straight line graphs are of the form y = mx + c 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑦 𝑤𝑖𝑡ℎ: 𝑣=4𝑥+0 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑥 𝑤𝑖𝑡ℎ:

Back to the distance time graph: What is the equation of the following? Rise: Run: All straight line graphs are of the form y = mx + c 𝑣=4𝑡+0 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑥 𝑤𝑖𝑡ℎ:

Back to the distance time graph: What is the equation of the following? Rise: Run: All straight line graphs are of the form y = mx + c 𝑇ℎ𝑖𝑠 𝑏𝑒𝑐𝑜𝑚𝑒𝑠: 𝑣=4𝑡+0 𝑣=4𝑡

What equation has both v & t in it? 𝑣=4𝑡 𝑣= 𝑑 𝑡 a= 𝑣 𝑡

Stretching Spring Experiment