Direct, inverse and quadratic relationship

 

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Some relationships studied in Physics exist between experimental values.

For example, in a previous lab you studied the relationship between the circumference of round objects and their diameter. The independent variable (the X value) was the diameter. The dependant variable (the Y value) was the circumference. When you plotted circumference versus diameter you found out that the relation ship was linear. Y = Pi X. This means that X and Y are directly proportional. You multiply any diameter (X) by PI to get the circumference (Y).

In Physics, the relationship between the experimental values (circumference and diameter) is shown using a technique called graphing.

The independent variable is always plotted along the x axis of the graph, and the dependent value is plotted along the y axis of the graph.

For another lab you plotted measured area (Y ) of different circles versus their radius (X). The graph showed you that the relationship was quadratic. Y = PI X² (the graph is a parabola).

In that case, Y (the area) varies as the square of X (the radius).

For many graphs, at a given point, you can compute the slope.

Slope = change in y / change in x

The slope tells by much Y increases when X increases by 1.

In Physics, the slope represent a physical quantity. For example:

 In a graph: displacement (m) vs time (s), the slope is the velocity (m/s).

Velocity (m/s)  vs time (s) , the slope is the acceleration (m/s/s)

Circumference (m)  vs diameter (m) , the slope is PI

In Physics, we also use the area under the graph. In calculus, you will learn how to compute these areas using integrals but for now, you just need to now how to compute the areas of triangle (0.5 b h) and rectangle (b h).

For example: The area under the graph velocity of an object  vs time elapsed is the displacement of the object during this time.

 

 

1) Observe the following table. X values are the input. Y values are the output values.

 

A) How do you go from X to Y ? Y = ___ X. (find the relationship between X and Y)

 

B) Plot the graph Y vs X. Find the slope. (rise over run)

 

 

C) The graph shows that Y is directly ______________ to X.

For every 1 unit increase in variable X, variable Y increases by __________ units.

2) Observe the following table. X values are the input. Y values are the output values.

 

A. Plot the graph Y vs X.

 

B. Find the slope. (it must be negative)

 

C. Find the relationship between X and Y. Y = _____________

Hint: Y = mX + B  you need to find m and B

 

D. For every 1 unit increase in variable X, variable decreases by __________

 

E.  the graph shows that as X increases , Y ____________________

3) Observe the table:

 

A. Find the relationship between X and Y. Y =

 

B. circle the right statement:

     Y increases as the square of X                  Y is proportional to X

 

C.  circle the right statement:

          If X doubles , Y doubles                If X doubles, Y quadruples

D. Plot Y vs X.

 

E. Is the slope constant ? Does the slope increases or decreases ?

 

F. The graph shows a : linear (a line ) or quadratic relationship (a parabola) ?

4)

Consider:  Y = 2 X²

A) Fill the table:

 

B) the graph is a _____________, and Y is proportional to the ____________ of X

 

C) the graph is a : line ? Parabola ? Hyperbola ?

 

5) X and Y can be inversely proportional.

If  Y = 12/X, Y and X are inversely proportional.

 

Consider Y = 12/X

 

A.  Fill the table:

 

B) Graph the relationship. Go by 2 for the X and Y axis.

 

C) When X increases, Y ______________. The graph is a : line or hyperbola ?

6) two quantities have an indirect squared proportion if an increase in one causes a squared decrease in the other.

 

Consider Y = 12/X²

 

A.  Fill the table

 

B. plot the relationship. (go by 2 on the axis).

C. As X increases , Y ____________

D. Y is inversely proportional to the squared of ________

In Physics, graphs are used to display and compare physical concepts.

Make sure you can identify a linear relationship  (line, y = m x) a quadratic relationship

(y = kx², parabola)  or a indirect relationship (y = 1/x, hyperbola)

 

7) Sometimes, you will need to compute the area under a graph to find a physical quantity.

For example, the area under a graph velocity vs time is the displacement.

 

Consider the graph:

 

Can you find the area under the graph ? x increases by 1 second (s) . y increases by 1 m/s