Question
The graph of the line mc003-1.jpg is transltated right 5 units and down 3 units. What is the equation of the new line?
Group of answer choices
mc003-2.jpg
mc003-3.jpg
mc003-4.jpg
mc003-5.jpg
There is no correct answer.
Group of answer choices
mc003-2.jpg
mc003-3.jpg
mc003-4.jpg
mc003-5.jpg
There is no correct answer.
Answers
Answer
3x to the power of 3 - 36x +81
Answered by
GPT-4o mini
To find the new equation after translating a line, we can follow these steps:
1. Let's denote the original equation of the line as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
2. Translating the line right by 5 units means we subtract 5 from the x-value. Thus, the transformation we'll apply to the equation is replacing \(x\) with \(x - 5\).
3. Translating the line down by 3 units means we subtract 3 from the y-value. Therefore, after substituting \(x\), we will also subtract 3 from the entire equation.
So, the new equation will look like this:
\[
y = m(x - 5) + b - 3
\]
This simplifies to:
\[
y = mx - 5m + b - 3
\]
We can combine the constant terms:
\(
new\_y\_intercept = b - 5m - 3
\)
Thus, the new equation of the line is:
\[
y = mx + (b - 5m - 3)
\]
Now, if you provide me with the original equation of the line from the first image (mc003-1.jpg), I can help you calculate the exact new equation. Please share that information.
1. Let's denote the original equation of the line as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
2. Translating the line right by 5 units means we subtract 5 from the x-value. Thus, the transformation we'll apply to the equation is replacing \(x\) with \(x - 5\).
3. Translating the line down by 3 units means we subtract 3 from the y-value. Therefore, after substituting \(x\), we will also subtract 3 from the entire equation.
So, the new equation will look like this:
\[
y = m(x - 5) + b - 3
\]
This simplifies to:
\[
y = mx - 5m + b - 3
\]
We can combine the constant terms:
\(
new\_y\_intercept = b - 5m - 3
\)
Thus, the new equation of the line is:
\[
y = mx + (b - 5m - 3)
\]
Now, if you provide me with the original equation of the line from the first image (mc003-1.jpg), I can help you calculate the exact new equation. Please share that information.
Answered by
GPT-4o mini
To find the new equation after translating the graph of \( y = 3x^3 - 36x + 81 \) right by 5 units and down by 3 units, we can follow these steps:
1. **Translation Right by 5 Units**: Replace \( x \) with \( x - 5 \).
\[
y = 3(x - 5)^3 - 36(x - 5) + 81
\]
2. **Expanding \( (x - 5)^3 \)**:
\[
(x - 5)^3 = x^3 - 15x^2 + 75x - 125
\]
Therefore,
\[
y = 3(x^3 - 15x^2 + 75x - 125) - 36(x - 5) + 81
\]
3. **Distributing the 3**:
\[
y = 3x^3 - 45x^2 + 225x - 375
\]
4. **Expanding \( -36(x - 5) \)**:
\[
-36(x - 5) = -36x + 180
\]
5. **Combining all the terms**:
\[
y = 3x^3 - 45x^2 + 225x - 375 - 36x + 180 + 81
\]
Simplifying further:
\[
y = 3x^3 - 45x^2 + (225x - 36x) + (-375 + 180 + 81)
\]
\[
y = 3x^3 - 45x^2 + 189x - 114
\]
6. **Now, translating down by 3 units**: Subtract 3 from the entire equation.
\[
y = 3x^3 - 45x^2 + 189x - 114 - 3
\]
\[
y = 3x^3 - 45x^2 + 189x - 117
\]
Thus, the new equation after the translation is:
\[
y = 3x^3 - 45x^2 + 189x - 117
\]
If you have options to choose from (like in the original question), you can compare this result to find the correct one.
1. **Translation Right by 5 Units**: Replace \( x \) with \( x - 5 \).
\[
y = 3(x - 5)^3 - 36(x - 5) + 81
\]
2. **Expanding \( (x - 5)^3 \)**:
\[
(x - 5)^3 = x^3 - 15x^2 + 75x - 125
\]
Therefore,
\[
y = 3(x^3 - 15x^2 + 75x - 125) - 36(x - 5) + 81
\]
3. **Distributing the 3**:
\[
y = 3x^3 - 45x^2 + 225x - 375
\]
4. **Expanding \( -36(x - 5) \)**:
\[
-36(x - 5) = -36x + 180
\]
5. **Combining all the terms**:
\[
y = 3x^3 - 45x^2 + 225x - 375 - 36x + 180 + 81
\]
Simplifying further:
\[
y = 3x^3 - 45x^2 + (225x - 36x) + (-375 + 180 + 81)
\]
\[
y = 3x^3 - 45x^2 + 189x - 114
\]
6. **Now, translating down by 3 units**: Subtract 3 from the entire equation.
\[
y = 3x^3 - 45x^2 + 189x - 114 - 3
\]
\[
y = 3x^3 - 45x^2 + 189x - 117
\]
Thus, the new equation after the translation is:
\[
y = 3x^3 - 45x^2 + 189x - 117
\]
If you have options to choose from (like in the original question), you can compare this result to find the correct one.
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