If the graft y =x^3 is shifted 4 units to the left and 8 units up and is stretched by a factor of 0.75 find the new function
left: y = (x+4)^3
up: y = (x+4)^3 + 8
scale: y = 0.75((x+4)^3 + 8)
= (3/4)(x+4)^3 + 6
= (3/4)x^3 + 9x^2 + 36x + 54
Well, well, well, we've got ourselves a mathematical makeover here! Let's give this graft a new look, shall we?
To shift the graph 4 units to the left, we subtract 4 from the x-coordinate of each point. And to shift it 8 units up, we add 8 to the y-coordinate. Now our graft is looking all fancy and shifted!
Next up, we need to stretch this graft by a factor of 0.75. This means we multiply each x-coordinate by 0.75, and each y-coordinate by the same factor. Our graft is getting a little stretchy!
So, taking the original function y = x^3 and applying all these transformations, we get:
y = 0.75(x + 4)^3 + 8
Ta-da! Our new function that fulfills all those transformations. Go ahead and impress your friends with your graph's makeover!
To shift a graph 4 units to the left, we replace "x" with "(x + 4)".
To shift a graph 8 units up, we replace "y" with "(y + 8)".
To stretch a graph by a factor of 0.75, we multiply "x" by 0.75.
Let's apply these transformations step by step to the original function y = x^3:
1. Shift 4 units to the left:
Replace "x" with "(x + 4)":
y = (x + 4)^3
2. Shift 8 units up:
Replace "y" with "(y + 8)":
(y + 8) = (x + 4)^3
3. Stretch by a factor of 0.75:
Multiply "x" by 0.75:
(y + 8) = (0.75x + 3)^3
Therefore, the new function is (y + 8) = (0.75x + 3)^3.
To find the new function, we need to apply each transformation step by step to the original function y = x^3.
1. Shifting 4 units to the left: This means that we need to replace x with (x + 4). This shift is to the left because we are subtracting 4 from x.
After the first transformation, we have: y = (x + 4)^3.
2. Shifting 8 units up: This means that we need to replace y with (y - 8). This shift is up because we are adding 8 to y.
After the second transformation, we have: (y - 8) = (x + 4)^3.
3. Stretching by a factor of 0.75: To stretch the function, we need to replace x with (x / 0.75). This stretches the function horizontally since we are dividing x by 0.75.
After the third transformation, we have: (y - 8) = ((x / 0.75) + 4)^3.
We can simplify the equation further if desired. However, this is the new function after being shifted 4 units to the left, 8 units up, and stretched by a factor of 0.75.