Complete the table assuming the type of change indicated.

Question

Complete the table assuming the type of change indicate. Then use the table to write the function.

a). Linear change b). Exponential change

x y x y

-3 12 -3 12

-2 -2

-1 3 -1 3

0 0

1 1

2 2

Answer 1

For each of these, you will use a line formula. For the Linear change, use the Point-slope form of a line. You are given the x and y coordinates of two points, P1(x₁, y₁) and P2(x₂, y₂) . It doesn't matter which point you call point 1 or which you call point 2, all that matters is that you are consistent

1) find the slope of the line: m = (y₂ - y₁)/( (x₂ - x₁)

2) use the slope found in 1) and one of the points. It doesn't matter which point you use as long as you use both x and y values from that point: y - y₁ = m (x - x₁) (this equation is also written: y - y₁ = m (x - x₁) + y₁

3) rearrange this equation so that you end up with this: y = mx + b, where b = y₁ - mx₁

4) substitute the value given for x in the table and solve for y.

For the exponential side, the basic form of any exponential function is y = a^x where a is a positive number not equal to 1. To find a, given y and x, take the log (or ln) of both sides of the equation.

This results in: log (y) = x log(b) OR ln(y) = x ln(b).

Divide through by x: (log(y) / x) = log(a), then 10^(log(y) / x) = a

OR (ln(y) / x) = ln(a), then e^(log(y) / x) = a

Both of these methods will give the same answer. It is important to remember - if you use log (base 10) use the 10, if you use ln (natural log) use e.

You can also apply the power 1/x to both sides of the equation; y^(1/x) = a^(x*(1/x)) = a

Any of these methods should work, and once you have a, then you can solve for the rest of the table. The problem is that the two points you have in your exponential table do not give the same value of a. Might want to verify the tables.

Answer 2

a) Two points on the line (-3, 12), (-1, 3) are given.

Use this two point, you can have a liner function.

Use your linear function, you can complete table in a).

b) Exp. function has form y = ab^x where b > 0.

Plug two points (-3, 12), (-1, 3) into the function.

Solve a and b, you get a exp function.

Use your exp function to finish the table in b).