Given the parent function f(x)log(base10)x, state the equation of the function that results from a vertical stretch by a factor of 2/5, a horizontal stretch by a factor of 3/4, a reflection in the y-axis , a horizontal translation 2 units to the right, and a vertical translation 3 units down.
I did this so far and I don't know what to write for a horizontal stretch by a factor of 3/4.
f(x)=alog(base10)(k(x-d))+c
=2/5log(base10)(-k(x-2))-3
Would it be 3/4 or different I know it says 1/k?
To incorporate a horizontal stretch by a factor of 3/4, we can think of it as a horizontal compression by a factor of 4/3. Therefore, the value of k should be 4/3.
The equation of the transformed function is:
f(x) = a * log(base10)[k(x - d)] + c
Given that it is a vertical stretch by a factor of 2/5, a reflection in the y-axis, a horizontal translation 2 units to the right, and a vertical translation 3 units down, we can substitute the values:
a = 2/5 (vertical stretch)
k = 4/3 (horizontal compression)
d = 2 (horizontal translation to the right)
c = -3 (vertical translation down)
Therefore, the equation becomes:
f(x) = (2/5) * log(base10)[(4/3)(x - 2)] - 3
To find the equation of the function after a horizontal stretch by a factor of 3/4, you are on the right track with the equation:
f(x) = a log(base10)(k(x-d)) + c
However, there is a small mistake in the current equation you have written. Instead of using "k" as the scaling factor for the horizontal stretch, you need to use "1/k" as per the equation.
So, the corrected equation for the horizontal stretch by a factor of 3/4 becomes:
f(x) = a log(base10)(1/(3/4)(x-d)) + c
Next, let's apply the other transformations mentioned in the question:
1. Vertical stretch by a factor of 2/5:
Multiply the entire function by 2/5:
f(x) = (2/5) * a log(base10)(1/(3/4)(x-d)) + (2/5) * c
2. Reflection in the y-axis:
Change the sign of the entire function:
f(x) = - (2/5) * a log(base10)(1/(3/4)(x-d)) - (2/5) * c
3. Horizontal translation 2 units to the right:
Replace x with (x - 2):
f(x) = - (2/5) * a log(base10)(1/(3/4)((x - 2) - d)) - (2/5) * c
4. Vertical translation 3 units down:
Subtract 3 from the function:
f(x) = - (2/5) * a log(base10)(1/(3/4)((x - 2) - d)) - (2/5) * c - 3
Now, this is the equation for the final function incorporating all the transformations. Note that the values of "a," "d," and "c" are not given in the question, so you will need to know those values to have a specific function.