Question
On a coordinate plane, point B(–6, 1) is translated to B prime(–3, –2). Indira uses these steps to find a rule to describe the translation.
Step 1 Substitute the original coordinates and the translated coordinates into (x, y) right-arrow (x + a, y + b):
B (negative 6, 1) right-arrow B prime (negative 6 + a, 1 + b) = B prime (negative 3, negative 2)
Step 2
Write two equations:
Negative 6 + a = negative 2. 1 + b = negative 3.
Step 3
Solve each equation:
Negative 6 + a = negative 2. a = negative 2 + 6. a = 4. 1 + b = 3. b = negative 3 minus 1. b = negative 4.
Step 4
Write the translation rule:
(x, y) right-arrow (x + 4, y minus 4)
Which corrects Indira’s first error?
Indira should have substituted B (negative 6, 1) right-arrow B prime (negative 3 + a, negative 2 + b) = B prime (negative 6, 1) in Step 1.
Indira should have written the equations Negative 6 + a = negative 3 and 1 + b = negative 2 in Step 2.
Indira should have solved the equations to find that a = negative 8 and b = negative 2 in Step 3.
Indira should have written the translation rule (x, y) right-arrow (x minus 4, y + 4) in Step 4.
Step 1 Substitute the original coordinates and the translated coordinates into (x, y) right-arrow (x + a, y + b):
B (negative 6, 1) right-arrow B prime (negative 6 + a, 1 + b) = B prime (negative 3, negative 2)
Step 2
Write two equations:
Negative 6 + a = negative 2. 1 + b = negative 3.
Step 3
Solve each equation:
Negative 6 + a = negative 2. a = negative 2 + 6. a = 4. 1 + b = 3. b = negative 3 minus 1. b = negative 4.
Step 4
Write the translation rule:
(x, y) right-arrow (x + 4, y minus 4)
Which corrects Indira’s first error?
Indira should have substituted B (negative 6, 1) right-arrow B prime (negative 3 + a, negative 2 + b) = B prime (negative 6, 1) in Step 1.
Indira should have written the equations Negative 6 + a = negative 3 and 1 + b = negative 2 in Step 2.
Indira should have solved the equations to find that a = negative 8 and b = negative 2 in Step 3.
Indira should have written the translation rule (x, y) right-arrow (x minus 4, y + 4) in Step 4.
Answers
GPT-4o mini
Let's go through Indira's steps to identify her error.
1. **Step 1** should establish the translation from B to B prime. The points are B(−6, 1) and B prime(−3, −2), so the translation rule should show that B is moved to B prime correctly.
In this case, since B(−6, 1) is translated to B prime(−3, −2), we can denote the translation in the form:
\[
B \rightarrow B' \text{ translates as } (x, y) \rightarrow (x + a, y + b)
\]
When substituting, the correct relation should be:
\[
(-6, 1) \rightarrow (-3, -2) \text{ gives } (-6 + a, 1 + b) = (-3, -2)
\]
Indira did this correctly.
2. **Step 2** involves setting up equations based on the coordinates after the translation. From the correct points:
- The translation in x-coordinates gives us:
\[
-6 + a = -3
\]
- The translation in y-coordinates gives us:
\[
1 + b = -2
\]
Indira wrote different equations:
- She incorrectly wrote \(-6 + a = -2\) (should be -3)
- She incorrectly wrote \(1 + b = -3\) (should be -2).
This means there is an error in how she wrote the equations.
3. **Step 3** identifies the solutions to the equations. Since she had the equations set up incorrectly, her calculations for 'a' and 'b' resulted in wrong values.
4. **Step 4** indicates the writing of the final translation rule. If her calculations were right, the rule would reflect those correct values, but that's irrelevant because the setup was wrong.
So, the correct choice addressing Indira's first error is:
**Indira should have written the equations \(-6 + a = -3\) and \(1 + b = -2\) in Step 2.**
1. **Step 1** should establish the translation from B to B prime. The points are B(−6, 1) and B prime(−3, −2), so the translation rule should show that B is moved to B prime correctly.
In this case, since B(−6, 1) is translated to B prime(−3, −2), we can denote the translation in the form:
\[
B \rightarrow B' \text{ translates as } (x, y) \rightarrow (x + a, y + b)
\]
When substituting, the correct relation should be:
\[
(-6, 1) \rightarrow (-3, -2) \text{ gives } (-6 + a, 1 + b) = (-3, -2)
\]
Indira did this correctly.
2. **Step 2** involves setting up equations based on the coordinates after the translation. From the correct points:
- The translation in x-coordinates gives us:
\[
-6 + a = -3
\]
- The translation in y-coordinates gives us:
\[
1 + b = -2
\]
Indira wrote different equations:
- She incorrectly wrote \(-6 + a = -2\) (should be -3)
- She incorrectly wrote \(1 + b = -3\) (should be -2).
This means there is an error in how she wrote the equations.
3. **Step 3** identifies the solutions to the equations. Since she had the equations set up incorrectly, her calculations for 'a' and 'b' resulted in wrong values.
4. **Step 4** indicates the writing of the final translation rule. If her calculations were right, the rule would reflect those correct values, but that's irrelevant because the setup was wrong.
So, the correct choice addressing Indira's first error is:
**Indira should have written the equations \(-6 + a = -3\) and \(1 + b = -2\) in Step 2.**