Write a rule for g that represents the indicated transformation of the graph of f.

Question

f(x)=e^x ; vertical shrink by a factor of 1/4, followed by a translation 5 units up

f(x)=log_(16)x; vertical stretch by a factor of 9, followed by translations 2 units right and 3 units down

Answer 1

Vertical shrinks or stretches come about when the function is multiplied by a "factor" or number. A stretch occurs when the number is greater than "1" and a shrink occurs when the number is larger than zero but less than "1".. Example: If the function is "x" (i.e., f(x) = x) then stretching it vertically by a factor of 4 means the function becomes f(x) = 4x. If the function is x² (i.e., f(x) = x²) then shrinking it vertically by a factor of 1/4 means the function becomes f(x) = 1/4x².

Vertical translations come about when a number is added or subtracted from a function. If a number is added to the function, it moves the graph up by that number. If a number is subtracted from the function, then it moves the graph down by that number. Example: If the function is f(x) = log(x) then to move the graph up 2 units, we make the function f(x) = log(x) + 2. To move the graph of f(x) = log(x) down by 6 units, we make the function f(x) = log(x) - 6.

Horizontal translations come about when a number is added or subtracted to the "x" BEFORE the function is performed. This actually is a little confusing because it is backwards from the way it "reads". If the function is f(x) = e^x then to move the graph up 2 units to the right, we actually subtract 2 from the "x" making the function f(x) = e^{(x - 2)}. Notice that the subtraction is "grouped with the "x". To move the graph of f(x) = e^x left by 5 units, we make the function f(x) = e^{(x + 5)}, again grouping the addition/subtraction with the "x" and using a "+" to move left (this case) or a "-" to move right.

So for f(x) = e^x; a vertical shrink by a factor of 1/4 followed by a translation 5 units up turns the function into g(x) = 1/4e^x + 5

And for f(x) = log₁₆(x) (if that is what you meant); a vertical stretch by a factor of 9 followed by translations 2 units right and 3 units down would turn make g(x) = 9log₁₆(x - 2) - 3