if f(x)= x+2, find the equation of the function F obtained by stretching the function f vertically by a factor of 2 and compressing the function horizontally by a factor of 2, followed by a translation of 1/2 horizontally and -3 vertically.
the equation is: -x^2 - 6x - 7
what happens to the points (0,2) and (-1,1) under the transformation? i don't know what to do/how to find the answer
Ah, transformations can be a bit tricky, but don't worry, I'm here to help! Let's break it down step by step.
First, we stretch the function f vertically by a factor of 2. This means that every y-coordinate of f(x) will be doubled. So, our new function F(x) can be written as F(x) = 2 * f(x).
Next, we compress the function horizontally by a factor of 2. This means that every x-coordinate of F(x) will be divided by 2. So, our new function F(x) becomes F(x) = 2 * f(x/2).
After that, we have a translation of 1/2 horizontally and -3 vertically. This means we shift the graph 1/2 units to the right and 3 units down. So, the new function F(x) will be F(x) = 2 * f((x/2) - 1/2) - 3.
Now that we have our transformed function, let's check what happens to the points (0,2) and (-1,1):
For the point (0,2):
- When we plug in x = 0 into F(x), we get F(0) = 2 * f((0/2) - 1/2) - 3.
- Simplifying, F(0) = 2 * f(-1/2) - 3.
- Using the original function f(x) = x + 2, we find that f(-1/2) = (-1/2) + 2 = 3/2.
- Substituting this back into F(0), we get F(0) = 2 * (3/2) - 3 = 0.
For the point (-1,1):
- When we plug in x = -1 into F(x), we get F(-1) = 2 * f((-1/2) - 1/2) - 3.
- Simplifying, F(-1) = 2 * f(-1) - 3.
- Using the original function f(x) = x + 2, we find that f(-1) = -1 + 2 = 1.
- Substituting this back into F(-1), we get F(-1) = 2 * 1 - 3 = -1.
So, under the given transformation, the point (0,2) becomes (0,0), and the point (-1,1) becomes (-1,-1).
To find the equation of the function F obtained by applying the given transformations to f(x) = x + 2, you can follow these steps:
Step 1: Vertical Stretching by a Factor of 2
Multiply the entire function by 2 to stretch it vertically. The new function after this transformation is 2f(x) = 2(x + 2).
Step 2: Horizontal Compression by a Factor of 2
Divide the input of the function by 2 to compress it horizontally. The new function after this transformation is 2f(x/2) = 2[(x/2) + 2].
Step 3: Horizontal Translation by 1/2
To shift the graph horizontally by 1/2, subtract 1/2 from the input of the function. The new function after this transformation is 2f[(x/2) - 1/2] = 2[((x/2) - 1/2) + 2].
Step 4: Vertical Translation by -3
To shift the graph vertically by -3, subtract 3 from the output of the function. The new function after this transformation is 2f[(x/2) - 1/2] - 3 = 2[((x/2) - 1/2) + 2] - 3.
Simplifying the equation, we get:
F(x) = 2[(x/2) + 3/2] - 3.
To find what happens to the points (0,2) and (-1,1) under the transformation, you need to substitute the x-coordinates of the points into the transformed equation and find the corresponding y-coordinates.
For the point (0,2):
Replacing x with 0, we get:
F(0) = 2[(0/2) + 3/2] - 3 = 2[0 + 3/2] - 3 = 2(3/2) - 3 = 3 - 3 = 0.
So, the transformed point is (0,0).
For the point (-1,1):
Replacing x with -1, we get:
F(-1) = 2[(-1/2) + 3/2] - 3 = 2[1/2] - 3 = 1 - 3 = -2.
So, the transformed point is (-1,-2).
Therefore, under the given transformations, the point (0,2) becomes (0,0) and the point (-1,1) becomes (-1,-2).
To find the equation of the function F obtained by the given transformations, we need to go step by step through each transformation and apply it to the original function f(x) = x + 2.
Step 1: Stretching Vertically by a Factor of 2
To stretch the function vertically by a factor of 2, we multiply the entire function by 2. So, the new function becomes F1(x) = 2 * f(x) = 2 * (x + 2).
F1(x) = 2x + 4
Step 2: Compressing Horizontally by a Factor of 2
To compress the function horizontally by a factor of 2, we divide the input (x) by 2. The function after this transformation is F2(x) = F1(x/2).
F2(x) = 2(x/2) + 4
F2(x) = x + 4
Step 3: Horizontal Translation by 1/2
To horizontally translate the function by 1/2, we subtract 1/2 from the input (x). The function after this transformation is F3(x) = F2(x - 1/2).
F3(x) = (x - 1/2) + 4
F3(x) = x + 3.5
Step 4: Vertical Translation by -3
To vertically translate the function down by 3 units, we subtract 3 from the function. The final function is F4(x) = F3(x) - 3.
F4(x) = (x + 3.5) - 3
F4(x) = x + 0.5
Therefore, the equation of the function F obtained by the given transformations is F(x) = x + 0.5.
Now, let's consider what happens to the points (0,2) and (-1,1) under these transformations.
For the point (0,2):
Original function: f(0) = 0 + 2 = 2
After transformation: F(0) = 0 + 0.5 = 0.5
So, under the given transformations, the point (0,2) is transformed to (0.5,2).
For the point (-1,1):
Original function: f(-1) = -1 + 2 = 1
After transformation: F(-1) = -1 + 0.5 = -0.5
So, under the given transformations, the point (-1,1) is transformed to (-0.5,1).
In summary, the point (0,2) is transformed to (0.5,2) and the point (-1,1) is transformed to (-0.5,1) under the given transformations.