The question was:
write the equation of the given parent graph with the points, (5,-2), (6,0),
The Bot determined that the equation of the line was y = 2x-12
That means the parent graph was probably y = x, but the graph had been dilated vertically by a factor of 2, and shifted down 12.
Yes, that's correct. The equation of the given parent graph is y = x. However, in this case, it has been vertically dilated by a factor of 2 and shifted downward by 12 units. Therefore, the equation of the transformed graph is y = 2x - 12.
To find the equation of the given parent graph, we need to determine the equation of the line that passes through the points (5, -2) and (6, 0).
Let's start by finding the slope (m) of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates of the two points:
m = (0 - (-2)) / (6 - 5)
m = (0 + 2) / 1
m = 2 / 1
m = 2
Now that we have the slope, we can plug it back into the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the coordinates (5, -2):
y - (-2) = 2(x - 5)
y + 2 = 2(x - 5)
Expanding the equation:
y + 2 = 2x - 10
To obtain the final form, we can simplify the equation by isolating y:
y = 2x - 10 - 2
y = 2x - 12
Therefore, the equation of the given parent graph, after dilation and vertical shift, is y = 2x - 12.
To find the equation of the given parent graph with the given points, we will first assume that the parent graph is y = x. We will then apply the necessary transformations to this equation based on the information provided.
Since the given points are (5,-2) and (6,0), we can substitute these values into the equation y = x to find the transformation factors.
For the point (5,-2):
-2 = 5 * a + b
For the point (6,0):
0 = 6 * a + b
Solving these two equations simultaneously will give us the values of a and b, which represent the dilation and translation factors of the parent graph.
Let's solve these equations:
From the second equation, we can rearrange it to isolate b:
b = -6 * a
Substituting this value of b into the first equation:
-2 = 5 * a + (-6 * a)
-2 = -a
a = 2
Now substituting the value of a back into the second equation to find b:
0 = 6 * 2 + b
0 = 12 + b
b = -12
Therefore, the dilation factor (a) is 2, and the translation factor (b) is -12.
We can now apply these transformation factors to the parent function y = x. The dilation factor 2 will vertically stretch the graph, and the translation factor -12 will shift the graph down by 12 units.
So, the equation of the given parent graph with the transformation factors is:
y = 2x - 12