helpp pleasee!

Question

how to find the rate of change?

Answer 1

The rate of change is a linear concept.  When a liner equation is graphed correctly, we have a visual which is a straight line.  We can determine the slope in 3 ways:  from the equation itself, from the graph (straight line) and from a set of points that have been graphed from plugging in a value (any value) for x and then calculating y to get (x,y) but we need at least 2 points to do that.

 

A.  We can also think of the rate of change as the result of calculation when we plug in a value for x.  In the linear equation x + 3 = y, plug in any value for x and then add 3 to graph the point.  So, if we  use 5 for x then add 3 we have a value of 8.  So, we graph the point at (5,8).  If we use 9 for x, then graph the point at (9,12) and so on.  We can clearly see that it is the value we choose for x that drives the placement of the point.  In other words, as the value of x increases by 1, then the y value of the graphed point also increases by 1.  The 3 in x +3 = y remains constant meaning that it does not determne the slope but it does determine the y-intercept (where the equation {line} crosses the y-axis.

 

The slope for the liner equation x + 3 = y is 1 because the x is multiplied by 1 for its identity.  The y-intercept is still 3.

The slope for 2x +3 = y is 2 because the value of each x is doubled so if we choose a value of 5 for x then 2(5) + 3 = y gives us a point of (5,13).  We can see that this slope is steeper (greater) than the slope of x + 3 = y.

The slope of 3x +3 = y is 3.

The slope of 4x + 3 = y is 4.

The slope of 1/2(x) +3 + y is 1/2, etc.

, 

 

The slope of x = y is 1 (because 1 times x is x.)

The slope of 2x = y is 2/

The slope of 3x = y is 3
The slope of x/2 = y is 1/2, etc.

 

So much for the easy way to determine the slope of a linear equation.  And it really is easy from the equation itself.

The graph of the equation (the line) requires a little more thought, but it is still easy.

 

B.  From the graph (straight line)

     B1.  Choose two points (from the line which is called the graph)  with x and y being whole numbers because the calculations are easier.  That means choose points where x and y are like (2,5) and (4,7) rather than (2.8, 5.8) and (4.36, 7.36).  You now have point A and Point B.

     B2.  Now imagine that these two points are the end points (vertices) of two perpendicular sides of a rectangle.

     B3.  Now draw a straight horizontal line from point A and a straight vertical line from point B so that they intersect at the vertex of a right angle.  We see that we have drawn half of a rectangle with the graph (the line of the equation) being the diagonal of the rectangle.

     B4.  Now count the number of horizontal units from point A to the vertex of the right angle.  This count is denominator of the slope.  
     B5.  Now count the number of vertical units from point B to the vertex of the right angle.  This count is the numerator of the slope.

     B6.  Simplify the ratio, i.e, the ratio 8/4 is 2 but the ration 8/5 remains 8/5.  The ration 3/6 is 1/2 but the ratio 7/3 remains 7/3.

     B7.  Think of slope as a hill.  If the slope is 9/2 That means that for every two feet of walking forward, we have 9 feet of going up.  That is a pretty drastic slope and one that is pretty much mountain climbing.  A slope of three would mean that for every forward foot horizontally, we would face a rise of three feet, which is exercise.  Think of a negative slope as gong down a hill.  

    B8.  We now reach the point where we think of slope as rise over run or the ratio that describes the rate of change.  Rate being a particular kind of ratio.

 

C.  We are also asked to determine slope when given a set of points from a graphed equation.  No, we will not see the equation.  No, we will not see the graph.  We will be givens the points, but this is still good because slope is rate of change and we can determine the amount of change from point A (x,y) to point B (x,y).  But remember, there are two changes - vertical/horizontal.

   C1.  We only need two points label them point A and point B.  Let's use point A is (3,7) and point B is (5,11).  The ordered pair for point A becomes x₁ and y becomes y_1.The ordered pair for point B becomes x₂ and y becomes y_2.

   C2.  Now we are going to subtract y₁ from y_2. so that we have 11 - 7.  This is the numerator of the ratio that is slope.

   C3.  Now we are going to subtract x₁ from x_x so that we have 5 - 3.  This is the denominator of the ratio that is slope.

   C4.  We now have (11 - 7) / (5 - 3).  Calculate to get 4/2 which is simplified to 2/1 or 2.  
   C5.  In the equation 2x + 1 which is the equation used to graph points A and B, we have a slope of 2.

 

So, here we have three ways to determine slope.  It is really easy if we are given  the linear equation.  If we are only given the graph (the line on graph paper), it is still pretty easy.  If we are given a set of points and asked to determine the slope, it is still pretty easy if we can keep our numbers from getting all mixed up. We need remember the slope formula which is y₂ - y_{1 /} x₂ - x₁

and again plugging the correct values in the right places.

 

Hope this helps. 

 

 

Answer 2

The rate of change is the slope of the line.