Question

Determine the type of function represented by the table. Explain your reasoning. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -3 & 1 & 5 & 9 & 13 \\ \hline y & 8 & -3 & -14 & -25 & -36 \\ \hline \end{array} $$

Short answer

The type of function represented by the table is a linear function. This is determined by the constant differences in the x-values and y-values.

Step-by-step solution

Recognize the Pattern

First thing is to observe the differences in x-values and y-values between consecutive points. Subtract each consecutive x and y and record the differences.

Calculate Differences

Differences in x-values are: 1 - (-3) = 4, 5 - 1 = 4, 9 - 5 = 4, 13 - 9 = 4. Differences in y-values are: -3 - 8 = -11, -14 - (-3) = -11, -25 - (-14) = -11, -36 - (-25) = -11.

Identify Function Type

As the differences in both x and y values are constant, this suggests a linear relationship. In other words, it is a linear function.

Key concepts

Function Types

Functions are essential in mathematics as they help describe relationships between two quantities. There are various types of functions, including linear, quadratic, and exponential functions. A linear function is one of the simplest forms and is represented by a straight line in a graph. It has a constant rate of change, which makes it easy to understand and predict values.

- **Linear Function:** This type focuses on a consistent relationship between variables, typically expressed as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.- **Quadratic Function:** Exhibiting a parabolic shape, these functions contain terms like \( ax^2 + bx + c \), adding complexity.- **Exponential Function:** These grow or decay quickly and are described by \( y = a \, b^x \), where \( b \) is the base.
Understanding these types helps in identifying the right pattern in data, crucial for applications in real-world problems.

Difference Calculation

Difference calculation is a technique used to analyze how variables change over a sequence. It's essential when determining the type of function you are working with. When calculating differences, consider:

- **Continuous Differences:** If the change between consecutive variables (such as x-values and y-values) is consistent, this often indicates a linear relationship. - **Variable Differences:** Changes that aren't consistent may signal other types like quadratic or exponential functions.
In the provided table: - Differences in x-values are consistently 4. - Differences in y-values are consistently -11.
This constant difference implies a consistent rate of change typical of linear functions. Calculating these differences is vital for identifying function types accurately.

Pattern Recognition

Recognizing patterns involves seeing consistent formations or structures within data. In mathematics, especially when working with functions, identifying patterns can guide you to the right function type. Key steps in pattern recognition include:

- **Look for Regularity:** In linear functions, the consistent increase or decrease in values hints at a uniform pattern. - **Identify Constant Changes:** As seen in our table, the change is constant: x increases by 4 and y decreases by 11 at each step.
By observing these consistent differences, we can predict subsequent values or even graph the function. Recognizing such regularity simplifies understanding and solving problems associated with functions.

Function Identification

Identifying a function involves determining what type of function a data set or graph represents. This is key in predicting future data points or understanding the relationship in a set of values. In steps to identify a function:

- **Examine Differences:** Start by calculating the differences in x and y-values to find any consistency, confirming a linear function in this case. - **Analyze Pattern:** If differences are constant, the function is likely linear, as seen from the table pattern.
The constant difference in both variables suggests a linear pattern. This method helps in quickly recognizing and classifying relationships, which is helpful in diverse scientific and engineering calculations. By understanding how to identify functions, solving mathematical problems becomes more intuitive and efficient.