Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Certainly! Let's carefully examine the given mappings to identify the underlying rule.
We have the following input-output pairs:
[tex]\[ \begin{array}{cccccc} 0 & 1 & 2 & 3 & 4 & 5 \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 2 & 5 & 8 & 11 & 14 & 17 \end{array} \][/tex]
To find the rule, we will analyze the relationship between each input [tex]\( x \)[/tex] and its corresponding output [tex]\( y \)[/tex].
1. Identify the change in mapping:
For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]
For [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
For [tex]\( x = 2 \)[/tex], [tex]\( y = 8 \)[/tex]
For [tex]\( x = 3 \)[/tex], [tex]\( y = 11 \)[/tex]
For [tex]\( x = 4 \)[/tex], [tex]\( y = 14 \)[/tex]
For [tex]\( x = 5 \)[/tex], [tex]\( y = 17 \)[/tex]
2. Examine the pattern:
Let's analyze the changes in the output values:
- [tex]\( 5 - 2 = 3 \)[/tex]
- [tex]\( 8 - 5 = 3 \)[/tex]
- [tex]\( 11 - 8 = 3 \)[/tex]
- [tex]\( 14 - 11 = 3 \)[/tex]
- [tex]\( 17 - 14 = 3 \)[/tex]
We see a consistent change of [tex]\( 3 \)[/tex] in the output when the input increases by [tex]\( 1 \)[/tex].
3. Determine the relationship:
Given the pattern, we can hypothesize that the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is linear: [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
From the pattern above, the slope [tex]\( m \)[/tex] is [tex]\( 3 \)[/tex] because the output increases by 3 for each increase of 1 in the input.
Let's determine the y-intercept [tex]\( c \)[/tex]:
Using the first pair [tex]\((0, 2)\)[/tex]:
[tex]\[ y = 3x + c \][/tex]
[tex]\[ 2 = 3(0) + c \][/tex]
[tex]\[ c = 2 \][/tex]
Thus, the rule is established as:
[tex]\[ y = 3x + 2 \][/tex]
4. Verify the rule:
To ensure our derived rule [tex]\( y = 3x + 2 \)[/tex] is correct, let's apply it to all given input values:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 3(0) + 2 = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 3(1) + 2 = 5 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 3(2) + 2 = 8 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 3(3) + 2 = 11 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 3(4) + 2 = 14 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 3(5) + 2 = 17 \)[/tex]
All the values match the given mapping perfectly.
Therefore, the rule for the given mapping is:
[tex]\[ y = 3x + 2 \][/tex]
We have the following input-output pairs:
[tex]\[ \begin{array}{cccccc} 0 & 1 & 2 & 3 & 4 & 5 \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 2 & 5 & 8 & 11 & 14 & 17 \end{array} \][/tex]
To find the rule, we will analyze the relationship between each input [tex]\( x \)[/tex] and its corresponding output [tex]\( y \)[/tex].
1. Identify the change in mapping:
For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]
For [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
For [tex]\( x = 2 \)[/tex], [tex]\( y = 8 \)[/tex]
For [tex]\( x = 3 \)[/tex], [tex]\( y = 11 \)[/tex]
For [tex]\( x = 4 \)[/tex], [tex]\( y = 14 \)[/tex]
For [tex]\( x = 5 \)[/tex], [tex]\( y = 17 \)[/tex]
2. Examine the pattern:
Let's analyze the changes in the output values:
- [tex]\( 5 - 2 = 3 \)[/tex]
- [tex]\( 8 - 5 = 3 \)[/tex]
- [tex]\( 11 - 8 = 3 \)[/tex]
- [tex]\( 14 - 11 = 3 \)[/tex]
- [tex]\( 17 - 14 = 3 \)[/tex]
We see a consistent change of [tex]\( 3 \)[/tex] in the output when the input increases by [tex]\( 1 \)[/tex].
3. Determine the relationship:
Given the pattern, we can hypothesize that the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is linear: [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
From the pattern above, the slope [tex]\( m \)[/tex] is [tex]\( 3 \)[/tex] because the output increases by 3 for each increase of 1 in the input.
Let's determine the y-intercept [tex]\( c \)[/tex]:
Using the first pair [tex]\((0, 2)\)[/tex]:
[tex]\[ y = 3x + c \][/tex]
[tex]\[ 2 = 3(0) + c \][/tex]
[tex]\[ c = 2 \][/tex]
Thus, the rule is established as:
[tex]\[ y = 3x + 2 \][/tex]
4. Verify the rule:
To ensure our derived rule [tex]\( y = 3x + 2 \)[/tex] is correct, let's apply it to all given input values:
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 3(0) + 2 = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 3(1) + 2 = 5 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 3(2) + 2 = 8 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 3(3) + 2 = 11 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 3(4) + 2 = 14 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 3(5) + 2 = 17 \)[/tex]
All the values match the given mapping perfectly.
Therefore, the rule for the given mapping is:
[tex]\[ y = 3x + 2 \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
