Math Problem Statement

Look at the table of x and y values. Write a linear (y = mx + b), quadratic (y = ax^2), or exponential (y = a(b)^x) function that models the data.

Solution

Let's analyze the table of values and determine the type of function that models the dataLet's analyze the table of values and determine the type of function that models the linear, quadratic, or exponential.

The data points are:

(1, 4) \\ (2, 2) \\ (3, 7) \\ (4, 14) \\ (5, 28) \\ \end{aligned}$$ ### 1. Checking for a linear function: For a linear function $$y = mx + b$$, the rate of change (slope) between consecutive points should be constant. Let's calculate the differences in the $$y$$-values: - From $$(1, 4)$$ to $$(2, 2)$$, the change in $$y$$ is $$2 - 4 = -2$$. - From $$(2, 2)$$ to $$(3, 7)$$, the change in $$y$$ is $$7 - 2 = 5$$. - From $$(3, 7)$$ to $$(4, 14)$$, the change in $$y$$ is $$14 - 7 = 7$$. - From $$(4, 14)$$ to $$(5, 28)$$, the change in $$y$$ is $$28 - 14 = 14$$. Since the differences are not constant, this is **not a linear function**. ### 2. Checking for a quadratic function: For a quadratic function $$y = ax^2 + bx + c$$, the second differences in the $$y$$-values should be constant. Let's compute the first and second differences: - First differences (as already calculated): - $$-2$$, $$5$$, $$7$$, $$14$$ - Second differences: - $$5 - (-2) = 7$$ -

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Linear Equations
Quadratic Equations
Exponential Equations

Formulas

Linear: y = mx + b
Quadratic: y = ax^2 + bx + c
Exponential: y = a(b)^x

Theorems

-

Suitable Grade Level

Grades 8-10