logarithmic function

Question

Given the parent function f(x)=log_1o 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 3/4, a reflection in the y-axis a horizontal translation 2 units to the right and a vertical translation 3 units down.

Answer 1

Start with vertical and horizontal translations first,

Vertical shifts deals with y values:

f(x) = log₁₀x - 3

//vertical translation 3 units down

Now the Horizontal translations (inside parenthesis with x opposite sign of transformation)

f(x) = log₁₀(x -2) - 3

//Horizontal translation 2 units to the right

Now that the translations are complete we do the stretches/compression's of a function

Let's start with the vertical stretch:

f(x) * (2/5) = log₁₀(x-2) - 3

//Notice that we apply the vertical factor to the left side because we are

impacting the y values with a vertical stretch

Now the Horizontal:

f(x) * (2/5) = log₁₀( (x - 2) * (3/4) ) - 3

We can simplify the argument in the parenthesis to make the problem simplified:

(x - 2) * (3/4)

(3/4)*x - (3/4) * 2

//distribute 3/4

(3/4)*x - (3/2)

// 3*2/4 can be simplified by reducing 2/4 to 1/2 making it 3/2

--> f(x) * (2/5) = log₁₀ ( (3/4)*x - (3/2) ) - 3

Now we can move the (2/5) to the other side to isolate the f(x):

f(x) = (5/2) * log₁₀ ( (3/4)*x - (3/2) ) - 3*(5/2)

//Dividing both sides by 2/5 is the same as multiplying

both sides by 5/2, we then distribute this value to

both terms on the right

Lastly we need to do the reflection about the y-axis:

f(x) = (5/2) * log₁₀ ( -(3/4)*x + (3/2) ) - (15/2)

//We invert the signs inside of the argument, +

becomes -, - becomes +

This now should be translated, stretched and flipped according to the problems specifications. Let me know if you have questions.