If is a line whose equation is y = 2x - 1, find the equation of the image of under each of the following translations:
a. (x, y) → (x, y - 2)
b. (x, y) → (x +3, y)
c. (x, y) → (x - 3, y + 2)
1,7
a. To find the equation of the image under the translation (x, y) → (x, y - 2), we need to substitute the new coordinates (x, y - 2) into the equation y = 2x - 1.
Substituting y - 2 for y in the equation, we get:
y - 2 = 2x - 1
Simplifying the equation, we get:
y = 2x - 1 + 2
y = 2x + 1
Therefore, the equation of the image of the line y = 2x - 1 under the translation (x, y) → (x, y - 2) is y = 2x + 1.
b. To find the equation of the image under the translation (x, y) → (x + 3, y), we need to substitute the new coordinates (x + 3, y) into the equation y = 2x - 1.
Substituting x + 3 for x in the equation, we get:
y = 2(x + 3) - 1
Simplifying the equation, we get:
y = 2x + 6 - 1
y = 2x + 5
Therefore, the equation of the image of the line y = 2x - 1 under the translation (x, y) → (x + 3, y) is y = 2x + 5.
c. To find the equation of the image under the translation (x, y) → (x - 3, y + 2), we need to substitute the new coordinates (x - 3, y + 2) into the equation y = 2x - 1.
Substituting y + 2 for y and x - 3 for x in the equation, we get:
y + 2 = 2(x - 3) - 1
Simplifying the equation, we get:
y + 2 = 2x - 6 - 1
y + 2 = 2x - 7
Therefore, the equation of the image of the line y = 2x - 1 under the translation (x, y) → (x - 3, y + 2) is y + 2 = 2x - 7.
To find the equation of the image of a line under a translation, we need to apply the given translation to each point on the original line and then determine the equation of the new line.
Let's start with the equation of the original line:
y = 2x - 1
a. Translation: (x, y) → (x, y - 2)
To apply this translation, we subtract 2 from the y-coordinate of each point on the original line.
So, for each point (x, y) on the original line, the corresponding point on the image line will have coordinates (x, y - 2).
Using this information, let's find the equation of the image line:
Substitute y - 2 for y in the equation of the original line:
y - 2 = 2x - 1
y = 2x - 1 + 2
y = 2x + 1
Therefore, the equation of the image line after this translation is: y = 2x + 1.
b. Translation: (x, y) → (x + 3, y)
To apply this translation, we add 3 to the x-coordinate of each point on the original line.
So, for each point (x, y) on the original line, the corresponding point on the image line will have coordinates (x + 3, y).
Using this information, let's find the equation of the image line:
Substitute x + 3 for x in the equation of the original line:
y = 2(x + 3) - 1
y = 2x + 6 - 1
y = 2x + 5
Therefore, the equation of the image line after this translation is: y = 2x + 5.
c. Translation: (x, y) → (x - 3, y + 2)
To apply this translation, we subtract 3 from the x-coordinate and add 2 to the y-coordinate of each point on the original line.
So, for each point (x, y) on the original line, the corresponding point on the image line will have coordinates (x - 3, y + 2).
Using this information, let's find the equation of the image line:
Substitute x - 3 for x and y + 2 for y in the equation of the original line:
y + 2 = 2(x - 3) - 1
y + 2 = 2x - 6 - 1
y = 2x - 7
Therefore, the equation of the image line after this translation is: y = 2x - 7.