Question
Two parabolas are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
The function graphed with a blue dotted line is translated to the function graphed as a solid orange line. If the same translation were applied to the solid orange function, what would be the coordinates of the new y-intercept (not pictured)?
(1 point)
The new y-intercept would be (
The function graphed with a blue dotted line is translated to the function graphed as a solid orange line. If the same translation were applied to the solid orange function, what would be the coordinates of the new y-intercept (not pictured)?
(1 point)
The new y-intercept would be (
Answers
GPT-4o mini
To determine the new y-intercept of the solid orange function after applying the same translation that moved the blue dotted line to the orange line, we need to first identify what type of translation occurred.
Let's assume the translation is represented by moving the graph right by \( h \) units and/or up by \( k \) units. The general translation of a function \( f(x) \) is given by:
\[
g(x) = f(x - h) + k
\]
This indicates that the y-intercept of the original function \( f(x) \) (where \( x = 0 \)) will shift to \( g(0) = f(-h) + k \).
1. **Identify the y-intercept of the blue dotted function** (let's say it is \( (0, b) \)).
2. **Apply the translation to find the new y-intercept of the orange function**, which would be \( (0, b + k) \) since the x-value does not change when we are finding the y-intercept.
3. **Translate the orange function again** using the same translation, leading to the final new y-intercept coordinates.
If, for instance, the blue parabola was translated 2 units up (k=2) and 1 unit right (h=1), then:
- The original blue y-intercept \( b \) would change to \( b + 2 \) for the orange function.
- If we apply the same translation (up 2 and right 1) to the orange function, its y-intercept would also increase by 2, resulting in:
\[
new \, y\text{-intercept} = (1, b + 4)
\]
If you still have the specific values for the y-intercept of the blue function or the translation values, you can calculate more precisely.
However, without those specific values, the final answer with the translation applied twice is:
The new y-intercept would be \( (h, b + 2k) \).
If we assume an example where initial y-intercept \( b = 0\) for simplicity, then the answer can be of the form \( (h, 4) \) if we considered \(k=2\) with h being a known value.
However, I need the exact translation values or the original y-intercept value to provide the specific coordinates. Let’s assume we find that \( b \) of the blue function was originally \( 0 \), we may examine the algebra derived above to conclude correct transformations.
Please clarify those points or provide the original function values to calculate. Otherwise, you'll simply fill in the specifics once they are gathered.
Let's assume the translation is represented by moving the graph right by \( h \) units and/or up by \( k \) units. The general translation of a function \( f(x) \) is given by:
\[
g(x) = f(x - h) + k
\]
This indicates that the y-intercept of the original function \( f(x) \) (where \( x = 0 \)) will shift to \( g(0) = f(-h) + k \).
1. **Identify the y-intercept of the blue dotted function** (let's say it is \( (0, b) \)).
2. **Apply the translation to find the new y-intercept of the orange function**, which would be \( (0, b + k) \) since the x-value does not change when we are finding the y-intercept.
3. **Translate the orange function again** using the same translation, leading to the final new y-intercept coordinates.
If, for instance, the blue parabola was translated 2 units up (k=2) and 1 unit right (h=1), then:
- The original blue y-intercept \( b \) would change to \( b + 2 \) for the orange function.
- If we apply the same translation (up 2 and right 1) to the orange function, its y-intercept would also increase by 2, resulting in:
\[
new \, y\text{-intercept} = (1, b + 4)
\]
If you still have the specific values for the y-intercept of the blue function or the translation values, you can calculate more precisely.
However, without those specific values, the final answer with the translation applied twice is:
The new y-intercept would be \( (h, b + 2k) \).
If we assume an example where initial y-intercept \( b = 0\) for simplicity, then the answer can be of the form \( (h, 4) \) if we considered \(k=2\) with h being a known value.
However, I need the exact translation values or the original y-intercept value to provide the specific coordinates. Let’s assume we find that \( b \) of the blue function was originally \( 0 \), we may examine the algebra derived above to conclude correct transformations.
Please clarify those points or provide the original function values to calculate. Otherwise, you'll simply fill in the specifics once they are gathered.
Two parabolas are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
GPT-4o mini
Based on your description, we have two parabolas on a coordinate plane, and the important aspect is determining the new y-intercept after repeating a translation on the solid orange line.
To answer the original question, let’s perform the following steps:
1. **Understand the Translation**:
- Without specific equations or additional details about the first (blue dotted) parabola's equation and its translation to the second (solid orange) parabola, we cannot provide the exact coordinates. However, if you could provide:
- The y-intercept of the blue parabola.
- The type of translation applied (how far it was moved horizontally or vertically).
2. **General Formulation of Translations**:
- If the blue parabola has a y-intercept at \( (0, b) \), and it was translated by \( h \) units horizontally and \( k \) units vertically, the new y-intercept of the orange parabola would be at:
\[
(0, b + k)
\]
- If you then apply the same translation again to the orange line, its y-intercept will shift to:
\[
(h, b + 2k)
\]
3. **Final New Y-Intercept**:
This means that the new y-intercept of the solid orange line, after the same translation has been applied to that function, will be at the point \( (h, b + 2k) \).
Since we need actual values of \( b \), \( h \), and \( k \), please provide the details about the:
- The original y-intercept of the blue dotted line.
- The specific translations that took place (how much to the right/left or up/down).
With that information, we can definitively calculate the new y-intercept coordinates accurately.
To answer the original question, let’s perform the following steps:
1. **Understand the Translation**:
- Without specific equations or additional details about the first (blue dotted) parabola's equation and its translation to the second (solid orange) parabola, we cannot provide the exact coordinates. However, if you could provide:
- The y-intercept of the blue parabola.
- The type of translation applied (how far it was moved horizontally or vertically).
2. **General Formulation of Translations**:
- If the blue parabola has a y-intercept at \( (0, b) \), and it was translated by \( h \) units horizontally and \( k \) units vertically, the new y-intercept of the orange parabola would be at:
\[
(0, b + k)
\]
- If you then apply the same translation again to the orange line, its y-intercept will shift to:
\[
(h, b + 2k)
\]
3. **Final New Y-Intercept**:
This means that the new y-intercept of the solid orange line, after the same translation has been applied to that function, will be at the point \( (h, b + 2k) \).
Since we need actual values of \( b \), \( h \), and \( k \), please provide the details about the:
- The original y-intercept of the blue dotted line.
- The specific translations that took place (how much to the right/left or up/down).
With that information, we can definitively calculate the new y-intercept coordinates accurately.