Pattern and Function - Functional Relationships in Tables and Graphs

Look at the following pattern in a table. If the Input ($x$) is $1, 2, 3, 4$ and the Output ($y$) is $7, 8, 9, 10$, find the rule and determine the output when the input is $15$.

Step 1: Compare each pair. $1 + 6 = 7$, $2 + 6 = 8$, $3 + 6 = 9$. The difference between $y$ and $x$ is always $6$. Step 2: Write the rule: $y = x + 6$. Step 3: Substitute $x = 15$ into the rule: $y = 15 + 6$. Step 4: Calculate the final result: $y = 21$.

This is an additive relationship. Since the output is consistently $6$ units higher than the input, we apply the rule $y = x + 6$ to find any unknown value.

A graph shows a line passing through the points $(1, 3)$, $(2, 6)$, and $(3, 9)$. What is the functional rule represented by this graph, and what would be the $y$ value if $x = 10$?

Step 1: Analyze the relationship between $x$ and $y$ for each point. For $(1, 3)$, $3 \div 1 = 3$. For $(2, 6)$, $6 \div 2 = 3$. For $(3, 9)$, $9 \div 3 = 3$. Step 2: Since the ratio is constant, the rule is multiplicative: $y = 3 \times x$. Step 3: To find the value at $x = 10$, calculate $y = 3 \times 10$. Step 4: $y = 30$.

By checking the coordinates on the graph, we see that $y$ is always triple the value of $x$. This identifies a multiplicative rule, which we then use to calculate the output for the given input.

$y = 4 \times x$ If $x = 20$: $y = 4 \times 20 = 80$ 80 matches are needed.

Comparing the input (squares) to the output (matches), we see that the output is always 4 times the input. This is a multiplicative relationship where the constant $k = 4$.

$y = x + 2$ On day 4 ($x = 4$): $y = 4 + 2 = 6 \text{ cm}$

Since the plant starts at 2 cm (the $y$-intercept when $x=0$) and increases by 1 each day, we add the number of days to the starting height. This is an additive relationship.