### Determining a possible equation for a curve when you are given two points.

Your first step should be to decide if you have a linear equation or an exponential equation (or neither).
• For a linear equation the slope between points should be constant: m =(y2 - y1)/(x2 / ab x1) = a constant
• For an exponential equation y = abx. In this case y2/y1 = abx2/abx1
• The a's cancel so the ratio of the y's is equal to the b coefficient to some power.
• If you use different points to solve for b you should get the same value of b each time.
• If b is a constant then it is reasonable to use an exponential function.

Linear equations: y = mx + b

1. If one of the points is on the y axis with coordinates (0, y)
1. b is equal to the given y value
2. find m by using its definition: M = (Y2- Y1)/(X2-X1)
3. now write the equation using the values of m and b.

Example:

• Put a linear equation through the points (0,9) and (2,15) b = 9 m = (15 - 9 )/(2 - 0) =6/2 = 3
• Final equation: y = 3x + 9
• If neither of the points are on the y axis.
1. calculate the slope, m, using the x,y coordinates of the two points.
2. (2) write the equation using the value of m that you just calculated
3. pick one of the points and substitute its x and y values, along with the value of m from step (1) into the basic equation, y = mx + b. You should have one unknown at this point, b. Solve for b.
4. Rewrite the equation using the values of m and b that you have calculated.

Example: Put a linear equation through the points (2,5) and (7, 11)

• m = (11-5)/(7-2) = 6/5 = (or) 1.2 if you prefer decimals
• y = 1.2x + b We will use point (2, 5)
• 5 = 1.2(2) + b So, b = 5 - 2.4 = 2.6
• Final equation: y = 1.2x + 2.6

Exponential equations: y = abx

1. If one of the points is on the y axis with coordinates (0, y)
• the value of a is equal to the value of the y coordinate of the point.
• use the (x,y) pair of the other point in the formula. The only unknown is b, so solve for it.
• rewrite the equation using the values for a and b that you just found.

Example: put an exponential equation through the points (0, 5) and (3, 40)

• The first point is on the y axis so the y coordinate 5 is equal to a 5 = ab0 b0 = 1 so 5 = a
• use the other point: 40 = 5b3 b3 = 40/5 = 8 This means that b is the cube root of 8 which is 2
• the formula is y = 5(2)x
• If neither of the points are on the y axis.
• take the ratio of the y values for the two points and set it equal to abx using the value of x for each point. This gives:
• the a's cancel and you are left with the ratio of the y's (which are numbers) equal to the ratio of b to two different powers. solve for b.
• rewrite the equation using the value of b you just calculated and the x and y values of one of the points. The only unknown in the equation will be a. Solve for a.
• write the equation of the function y = abx

Example: Put an exponential equation through the points (2, 45) and (4, 405) (1) take the ratio of the y values and solve for b (the a’s cancel) ? (2) rewrite the equation using your value of b and one of the points, we will use the point (2, 45) 45 = a(3)2 45 = a (9 a = 45/9 = 5 (3) write the final equation using the values of a and b found above y = 5(3)x