Certainly! Let's go through the steps to determine the input, the function, and the output from the table provided:
1. Identify the inputs and outputs:
- Inputs (x-values): [tex]\( x = -2, -1, 0, 1, 2 \)[/tex]
- Outputs (y-values): [tex]\( y = -5, -2, 1, 4, 7 \)[/tex]
2. Determine the relationship (function) between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- We need to find a function [tex]\( y = f(x) \)[/tex] that correctly maps each [tex]\( x \)[/tex] to its corresponding [tex]\( y \)[/tex].
- Observing the changes in [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases, we see that [tex]\( y \)[/tex] increments by 3 each time [tex]\( x \)[/tex] increases by 1. This suggests a linear relationship of the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
3. Find the slope [tex]\( m \)[/tex]:
- The slope [tex]\( m \)[/tex] can be determined by noticing the change in [tex]\( y \)[/tex] values per unit change in [tex]\( x \)[/tex].
- Here, the difference [tex]\( \Delta y \)[/tex] for each unit increase in [tex]\( x \)[/tex]:
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\( y \)[/tex] changes from -5 to -2, a change of [tex]\( -2 - (-5) = 3 \)[/tex].
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( y \)[/tex] changes from -2 to 1, a change of [tex]\( 1 - (-2) = 3 \)[/tex].
- Similarly, the change is consistent at 3 for other values.
- Thus, the slope [tex]\( m = 3 \)[/tex].
4. Find the y-intercept [tex]\( c \)[/tex]:
- Using one of the points, let's pick the point where [tex]\( x = 0 \)[/tex] and [tex]\( y = 1 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] and [tex]\( y = 1 \)[/tex] into the linear equation [tex]\( y = 3x + c \)[/tex]:
[tex]\[
1 = 3(0) + c \implies c = 1
\][/tex]
5. Write the function:
- Combining the slope and y-intercept, we get:
[tex]\[
y = 3x + 1
\][/tex]
6. Verify the function with all given points:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 3(-2) + 1 = -6 + 1 = -5
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 3(-1) + 1 = -3 + 1 = -2
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 3(0) + 1 = 0 + 1 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 3(1) + 1 = 3 + 1 = 4
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 3(2) + 1 = 6 + 1 = 7
\][/tex]
All calculations match the output values in the given table.
Summary:
- Input values: [tex]\( x = -2, -1, 0, 1, 2 \)[/tex]
- Determined function: [tex]\( y = 3x + 1 \)[/tex]
- Output values: [tex]\( y = -5, -2, 1, 4, 7 \)[/tex]
So, the function that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in this table is [tex]\( y = 3x + 1 \)[/tex].
