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QuestionDetermine the type of function for the points: (5, -1), (6, 6), (7, 25), (8, 62), (9, 123).

Studdy Solution

STEP 1

Assumptions1. We have a function defined by a set of (x,y)(x, y) pairs. . We are trying to determine if the function is linear, quadratic, cubic, or exponential.

STEP 2

First, let's look at the differences between consecutive yy values. If the function is linear, these differences should be constant.
Δy=yn+1−yn\Delta y = y_{n+1} - y_n

STEP 3

Calculate the differences between consecutive yy values.
Δy=[6−(−1),25−6,62−25,123−62]=[7,19,37,61]\Delta y = [6 - (-1),25 -6,62 -25,123 -62] = [7,19,37,61]

STEP 4

The differences between consecutive yy values are not constant, so the function is not linear. Now, let's look at the differences of the differences. If the function is quadratic, these should be constant.
Δ2y=Δyn+1−Δyn\Delta^2 y = \Delta y_{n+1} - \Delta y_n

STEP 5

Calculate the differences of the differences.
Δ2y=[19−7,37−19,61−37]=[12,18,24]\Delta^2 y = [19 -7,37 -19,61 -37] = [12,18,24]

STEP 6

The differences of the differences are not constant, so the function is not quadratic. Now, let's look at the differences of the differences of the differences. If the function is cubic, these should be constant.
Δ3y=Δ2yn+1−Δ2yn\Delta^3 y = \Delta^2 y_{n+1} - \Delta^2 y_n

STEP 7

Calculate the differences of the differences of the differences.
Δ3y=[18−12,24−18]=[6,6]\Delta^3 y = [18 -12,24 -18] = [6,6]

STEP 8

The differences of the differences of the differences are constant, so the function is cubic.
The function is cubic.

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