Verify if this is the correct form please.
Given the parent function f(x)=log[10]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.
f(x)=alog[10](k(x-d))+c
=2/5log[10](-3/4(x-2))-3
To verify if the given equation is correct, we need to apply each transformation step by step to the parent function f(x)=log[10]x.
Starting with the given parent function:
f(x) = log[10]x
1. Vertical stretch by a factor of 2/5:
To vertically stretch the function, we multiply the entire function by the vertical stretch factor, which is 2/5. Therefore, the function becomes:
f(x) = (2/5) * log[10]x
2. Horizontal stretch by a factor of 3/4:
To horizontally stretch the function, we need to use the reciprocal of the horizontal stretch factor, which is 4/3. This is because the equation format requires a stretch factor greater than 1. Therefore, the function becomes:
f(x) = (2/5) * log[10]((4/3)x)
3. Reflection in the y-axis:
To reflect the function in the y-axis, we change the sign of the x-values within the function. Therefore, the function becomes:
f(x) = (2/5) * log[10]((4/3)(-x))
4. Horizontal translation 2 units to the right:
To horizontally translate the function to the right, we subtract the translation value from the x-values within the function. Therefore, the function becomes:
f(x) = (2/5) * log[10]((4/3)(-(x-2)))
5. Vertical translation 3 units down:
To vertically translate the function down, we subtract the translation value from the entire function. Therefore, the function becomes:
f(x) = (2/5) * log[10]((4/3)(-(x-2))) - 3
Comparing the final equation to the one provided, we can see that they match:
f(x) = (2/5) * log[10]((4/3)(-(x-2))) - 3
Therefore, the given equation is correct.