Question

$$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 3 & 6 & 9 & 12 & 15 \\ \hline \boldsymbol{y} & 0.25 & 1 & 4 & 16 & 64 & 256 \\ \hline \end{array} $$$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -4 & -3 & -2 & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{y} & 16 & 8 & 4 & 2 & 1 & \frac{1}{2} & \frac{1}{4} \\ \hline \end{array} $$

Short answer

The relationship that defines y in terms of x is \( y = 0.25*4^x \) for the first table and \( y = 2^{-x} \) for the second table.

Step-by-step solution

Identify a Pattern

Observe the values of x and y in both tables. On increasing x by 3 in the first table, the y value is multiplied by 4, leading to the hypothesis that y could follow an exponential function of \( y = 4^x \). Similarly, for the second table, on increasing the x by 1 the y value is halved. Hence the hypothesized function is \( y = 2^{-x} \)

Verify the Function

Substitute a few x values from the tables into the hypothesized functions and check if they yield the corresponding y values. For example, in the first table if x=3, y should be \( 4^1 = 4 \), which matches the data. In the second table, if x=-3, y should be \( 2^{-(-3)} = 8 \), which also matches. So the hypotheses are correct.

Solve for a in each function

There might be a scaling factor 'a' in the function \( a*4^x \) or \( a*2^{-x} \). We can solve for 'a' by substituting x=0 into each function, because for any number except 0, any power of 0 is 1. So if we substitute x=0 in the first table we get \( a*4^0 = 0.25 \) and hence a=0.25. Similarly for the second table on substituting 0, we get \( a*2^{-0} = 1 \) and hence a=1. So the functions for each table are \( y = 0.25*4^x \) and \( y = 2^{-x} \).

Key concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra, encompassing numbers, variables, and arithmetic operations. They enable us to represent patterns and relationships between quantities abstractly. In the context of our exercise, algebraic expressions manifest as functions that model what happens to the variable 'y' as 'x' changes. The expressions are in the form of exponential functions, specifically:\[ y = a \times 4^x \] and \[ y = a \times 2^{-x} \].

These algebraic expressions are powerful as they compactly encapsulate the behavior of the data in the tables. Notice how an understanding of algebraic notation helps bridge the gap between the numerical patterns and the symbolic representation of those patterns. This deepens our understanding of mathematical concepts and assists in solving more complex problems that involve exponential growth or decay.

Patterns in Mathematics

In mathematics, patterns serve as blueprints for predicting future outcomes and unraveling the properties of numbers. They are the rhythmic guides that lead to the discovery of mathematical rules and formulas. The exercise presented here showcases a clear pattern in the relationship between 'x' and 'y'.

For the first set of data, we observe that as 'x' increases by 3, 'y' is multiplied by 4. This points to an exponential pattern, where 'y' is an exponential function of 'x'. Conversely, the second set of data reveals that with each increment of 1 in 'x', 'y' is reduced to half. This suggests an inverse exponential relationship.

Understanding and identifying these patterns is a fundamental skill in mathematics that not only helps in formulating equations but also in predicting and extending sequences. Recognizing a pattern early in a problem-solving process can often lead to a swift and elegant solution.

Functions in Algebra

Functions in algebra are like machines: you input a number, the function processes it following a set of rules, and then outputs a corresponding value. They are essential tools for modeling real-world scenarios and exploring the relationship between variables. In our exercise, we deal specifically with exponential functions. These functions are characterized by an equation of the form:\[ f(x) = ab^x \],

where 'a' is the initial amount, 'b' is the base or the growth (or decay) factor, and 'x' is the exponent indicating the power to which the base is raised.

Understanding Exponential Growth and Decay

Exponential growth occurs when the base 'b' is greater than 1, resulting in increasing y-values as x increases. Conversely, exponential decay transpires when 0 < 'b' < 1, leading to decreasing y-values with increasing x. This concept can be visualized in the graphs of our functions and is crucial for topics ranging from compound interest in finance to population dynamics in biology.