Question
Use input-output tables to determine which graph accurately represents the function f(x)=12x+3 with inputs of −4 , −2 , 0, 2, and 4.
Responses
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Responses
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Answers
GPT-4o mini
To determine which graph accurately represents the function \( f(x) = 12x + 3 \) with inputs of \( -4, -2, 0, 2, \) and \( 4 \), let's first calculate the corresponding output values for each input using the function.
1. **For \( x = -4 \)**:
\[
f(-4) = 12(-4) + 3 = -48 + 3 = -45
\]
2. **For \( x = -2 \)**:
\[
f(-2) = 12(-2) + 3 = -24 + 3 = -21
\]
3. **For \( x = 0 \)**:
\[
f(0) = 12(0) + 3 = 0 + 3 = 3
\]
4. **For \( x = 2 \)**:
\[
f(2) = 12(2) + 3 = 24 + 3 = 27
\]
5. **For \( x = 4 \)**:
\[
f(4) = 12(4) + 3 = 48 + 3 = 51
\]
Now we have the following points derived from the function:
- \( (-4, -45) \)
- \( (-2, -21) \)
- \( (0, 3) \)
- \( (2, 27) \)
- \( (4, 51) \)
Next, let’s check the options in the images provided:
- We need to find a graph that includes the points calculated above, particularly for negative values and outputs that go significantly below zero and also above 3.
Based on this:
- The graph should have a range on the x-axis from at least \( -4 \) to \( 4 \).
- The y-axis should extend significantly below zero (since \( -45 \) and \( -21 \) are lower than -10).
Considering this information, the second option appears to match the requirements. This graph:
- Has the x-axis ranging from \( -4 \) to \( 4 \).
- Has the y-axis ranging from \( -10 \) to \( 10 \), but it does not extend enough to cover \( -45 \) and \( -21 \). Therefore, we should verify if this option covers the aspect of a linear function passing through these extreme outputs.
Since outputs like \( -45 \) and \( 51 \) are not represented using the given ranges in the imagery, none of the options completely align due to the necessity of exceeding the \( -10 \) and likely needing to adjust upwards into higher values.
Ultimately, choose a graph with \( x \) range that contains \( -4 \) to \( 4 \), and ideally, the values should be depicted well above \( 3 \) and reflect the drastic rise in outputs for positive \( x \). Verify the required setting that shows large negative and significantly positive points as best aligning with the linear nature of the function and its extreme outputs. The second option, with closer x-axis alignment and extreme points for better calculations, is a reasonable match.
1. **For \( x = -4 \)**:
\[
f(-4) = 12(-4) + 3 = -48 + 3 = -45
\]
2. **For \( x = -2 \)**:
\[
f(-2) = 12(-2) + 3 = -24 + 3 = -21
\]
3. **For \( x = 0 \)**:
\[
f(0) = 12(0) + 3 = 0 + 3 = 3
\]
4. **For \( x = 2 \)**:
\[
f(2) = 12(2) + 3 = 24 + 3 = 27
\]
5. **For \( x = 4 \)**:
\[
f(4) = 12(4) + 3 = 48 + 3 = 51
\]
Now we have the following points derived from the function:
- \( (-4, -45) \)
- \( (-2, -21) \)
- \( (0, 3) \)
- \( (2, 27) \)
- \( (4, 51) \)
Next, let’s check the options in the images provided:
- We need to find a graph that includes the points calculated above, particularly for negative values and outputs that go significantly below zero and also above 3.
Based on this:
- The graph should have a range on the x-axis from at least \( -4 \) to \( 4 \).
- The y-axis should extend significantly below zero (since \( -45 \) and \( -21 \) are lower than -10).
Considering this information, the second option appears to match the requirements. This graph:
- Has the x-axis ranging from \( -4 \) to \( 4 \).
- Has the y-axis ranging from \( -10 \) to \( 10 \), but it does not extend enough to cover \( -45 \) and \( -21 \). Therefore, we should verify if this option covers the aspect of a linear function passing through these extreme outputs.
Since outputs like \( -45 \) and \( 51 \) are not represented using the given ranges in the imagery, none of the options completely align due to the necessity of exceeding the \( -10 \) and likely needing to adjust upwards into higher values.
Ultimately, choose a graph with \( x \) range that contains \( -4 \) to \( 4 \), and ideally, the values should be depicted well above \( 3 \) and reflect the drastic rise in outputs for positive \( x \). Verify the required setting that shows large negative and significantly positive points as best aligning with the linear nature of the function and its extreme outputs. The second option, with closer x-axis alignment and extreme points for better calculations, is a reasonable match.