Question
Write an equation of the graph in its final position.
The graph of y equals 4 Superscript xy=4x is translated 88 units to the left and then 22 units upward.
Question content area bottom
Part 1
Which of the following is the equation of the graph?
A.y equals 4 Superscript x plus 8 Baseline minus 2
y equals 4 Superscript x plus 8 Baseline minus 2y=4x+8−2
B.y equals 4 Superscript x minus 8 Baseline plus 2
y equals 4 Superscript x minus 8 Baseline plus 2y=4x−8+2
C.y equals 4 Superscript x minus 8 Baseline minus 2
y equals 4 Superscript x minus 8 Baseline minus 2y=4x−8−2
D.y equals 4 Superscript x plus 8 Baseline plus 2
y equals 4 Superscript x plus 8 Baseline plus 2y=4x+8+2
The graph of y equals 4 Superscript xy=4x is translated 88 units to the left and then 22 units upward.
Question content area bottom
Part 1
Which of the following is the equation of the graph?
A.y equals 4 Superscript x plus 8 Baseline minus 2
y equals 4 Superscript x plus 8 Baseline minus 2y=4x+8−2
B.y equals 4 Superscript x minus 8 Baseline plus 2
y equals 4 Superscript x minus 8 Baseline plus 2y=4x−8+2
C.y equals 4 Superscript x minus 8 Baseline minus 2
y equals 4 Superscript x minus 8 Baseline minus 2y=4x−8−2
D.y equals 4 Superscript x plus 8 Baseline plus 2
y equals 4 Superscript x plus 8 Baseline plus 2y=4x+8+2
Answers
Answered by
GPT-4o mini
To find the equation of the graph after the given translations, we need to apply the transformations step-by-step to the original equation \( y = 4^x \).
1. **Translation 88 units to the left:**
Translating a graph to the left affects the \( x \)-coordinate in the equation. The general transformation for a left translation of \( d \) units is \( f(x) \to f(x + d) \). In this case, \( d = 88 \), so we have:
\[
y = 4^{(x + 88)}
\]
2. **Translation 22 units upward:**
Translating a graph upward affects the \( y \)-coordinate by adding the amount to the output value of the function. So, for an upward translation of \( k \) units, the transformation is \( f(x) \to f(x) + k \). Here, \( k = 22 \), so we have:
\[
y = 4^{(x + 88)} + 22
\]
Now we can simplify the equation slightly.
However, in your answer choices, it appears that the transformed equation has been shifted into a different format using the properties of exponents.
Rewriting \( 4^{(x + 88)} \):
\[
= 4^x \cdot 4^{88}
\]
So:
\[
y = 4^x \cdot 4^{88} + 22
\]
This shows that the base function is still maintained, and the 22 is simply added.
Now, according to the options you've provided:
- **A:** \( y = 4^{(x + 8)} - 2 \)
- **B:** \( y = 4^{(x - 8)} + 2 \)
- **C:** \( y = 4^{(x - 8)} - 2 \)
- **D:** \( y = 4^{(x + 8)} + 2 \)
It looks like none of the direct forms we derived matches exactly, but given the expected form in the options, we would check again:
Recall, the significant translations were incorrectly applied to values supposed to maintain the constant; hence, upon realizing that:
- \( 88 \) units left translates to an \( x + 88 \).
- Anticipating the vertical upward addition by \( 22 \).
Correctly mapping values through your choices directly indicates focusing primarily on graphical distinctiveness (which could mask true shifts).
Thus, the closest equation conforming to upwards shift would correspond to **D**. Since all other equations either under-correct or mis-represent terms by divisions in signs.
In conclusion, the final equation modeled unto an implicit translation context aligns most passive with:
**Answer: D** \( y = 4^{(x + 8)} + 2 \)
Given translation misalignment feedbacks on simplifications.
1. **Translation 88 units to the left:**
Translating a graph to the left affects the \( x \)-coordinate in the equation. The general transformation for a left translation of \( d \) units is \( f(x) \to f(x + d) \). In this case, \( d = 88 \), so we have:
\[
y = 4^{(x + 88)}
\]
2. **Translation 22 units upward:**
Translating a graph upward affects the \( y \)-coordinate by adding the amount to the output value of the function. So, for an upward translation of \( k \) units, the transformation is \( f(x) \to f(x) + k \). Here, \( k = 22 \), so we have:
\[
y = 4^{(x + 88)} + 22
\]
Now we can simplify the equation slightly.
However, in your answer choices, it appears that the transformed equation has been shifted into a different format using the properties of exponents.
Rewriting \( 4^{(x + 88)} \):
\[
= 4^x \cdot 4^{88}
\]
So:
\[
y = 4^x \cdot 4^{88} + 22
\]
This shows that the base function is still maintained, and the 22 is simply added.
Now, according to the options you've provided:
- **A:** \( y = 4^{(x + 8)} - 2 \)
- **B:** \( y = 4^{(x - 8)} + 2 \)
- **C:** \( y = 4^{(x - 8)} - 2 \)
- **D:** \( y = 4^{(x + 8)} + 2 \)
It looks like none of the direct forms we derived matches exactly, but given the expected form in the options, we would check again:
Recall, the significant translations were incorrectly applied to values supposed to maintain the constant; hence, upon realizing that:
- \( 88 \) units left translates to an \( x + 88 \).
- Anticipating the vertical upward addition by \( 22 \).
Correctly mapping values through your choices directly indicates focusing primarily on graphical distinctiveness (which could mask true shifts).
Thus, the closest equation conforming to upwards shift would correspond to **D**. Since all other equations either under-correct or mis-represent terms by divisions in signs.
In conclusion, the final equation modeled unto an implicit translation context aligns most passive with:
**Answer: D** \( y = 4^{(x + 8)} + 2 \)
Given translation misalignment feedbacks on simplifications.