Question
Compare the function 3x+2y=12
to the function graphed in the following image to identify the true statement.
graph shows a line passing through (0,4) and (4,0)
(1 point)
Responses
The two functions have the same y-intercept.
The two functions have the same y-intercept.
The two functions have the same x-intercept.
The two functions have the same x-intercept.
The y-intercept of the graphed function is greater than the y-intercept of the function 3x + 2y = 12.
The y-intercept of the graphed function is greater than the y-intercept of the function 3x + 2y = 12.
The x-intercept of the graphed function is greater than the x-intercept of the function 3x + 2y = 12.
The x-intercept of the graphed function is greater than the x-intercept of the function 3x + 2y = 12.
to the function graphed in the following image to identify the true statement.
graph shows a line passing through (0,4) and (4,0)
(1 point)
Responses
The two functions have the same y-intercept.
The two functions have the same y-intercept.
The two functions have the same x-intercept.
The two functions have the same x-intercept.
The y-intercept of the graphed function is greater than the y-intercept of the function 3x + 2y = 12.
The y-intercept of the graphed function is greater than the y-intercept of the function 3x + 2y = 12.
The x-intercept of the graphed function is greater than the x-intercept of the function 3x + 2y = 12.
The x-intercept of the graphed function is greater than the x-intercept of the function 3x + 2y = 12.
Answers
GPT-4o mini
To analyze the given equation \(3x + 2y = 12\) and compare it to the function represented in the graph, let's first derive the y-intercept and x-intercept of the equation.
1. **Finding the y-intercept of the equation \(3x + 2y = 12\):**
- Set \(x = 0\):
\[
3(0) + 2y = 12 \implies 2y = 12 \implies y = 6
\]
- The y-intercept is 6, which corresponds to the point \((0, 6)\).
2. **Finding the x-intercept of the equation \(3x + 2y = 12\):**
- Set \(y = 0\):
\[
3x + 2(0) = 12 \implies 3x = 12 \implies x = 4
\]
- The x-intercept is 4, which corresponds to the point \((4, 0)\).
3. **Interpreting the graph provided:**
- The graph shows a line passing through the points \((0, 4)\) and \((4, 0)\).
- From this, we can see that the y-intercept of the graph is 4 (the point where the line crosses the y-axis) and the x-intercept is also 4 (the point where the line crosses the x-axis).
4. **Comparing intercepts:**
- The y-intercept of the function \(3x + 2y = 12\) is 6, while the y-intercept of the graphed function is 4. Therefore, the statement regarding the y-intercepts is:
- The y-intercept of the graphed function is **not greater than** the y-intercept of the function \(3x + 2y = 12\).
- The x-intercept of both functions is 4. Therefore, the statement regarding the x-intercepts is:
- The x-intercept of both functions is **the same** (4).
Based on this analysis, the correct conclusion is:
**The two functions have the same x-intercept.**
1. **Finding the y-intercept of the equation \(3x + 2y = 12\):**
- Set \(x = 0\):
\[
3(0) + 2y = 12 \implies 2y = 12 \implies y = 6
\]
- The y-intercept is 6, which corresponds to the point \((0, 6)\).
2. **Finding the x-intercept of the equation \(3x + 2y = 12\):**
- Set \(y = 0\):
\[
3x + 2(0) = 12 \implies 3x = 12 \implies x = 4
\]
- The x-intercept is 4, which corresponds to the point \((4, 0)\).
3. **Interpreting the graph provided:**
- The graph shows a line passing through the points \((0, 4)\) and \((4, 0)\).
- From this, we can see that the y-intercept of the graph is 4 (the point where the line crosses the y-axis) and the x-intercept is also 4 (the point where the line crosses the x-axis).
4. **Comparing intercepts:**
- The y-intercept of the function \(3x + 2y = 12\) is 6, while the y-intercept of the graphed function is 4. Therefore, the statement regarding the y-intercepts is:
- The y-intercept of the graphed function is **not greater than** the y-intercept of the function \(3x + 2y = 12\).
- The x-intercept of both functions is 4. Therefore, the statement regarding the x-intercepts is:
- The x-intercept of both functions is **the same** (4).
Based on this analysis, the correct conclusion is:
**The two functions have the same x-intercept.**