What is the rule for g(x)?

Question

Let h(x)=2x+4. If g(x) is a vertical stretch of 3 units and a horizontal translation 6 units left, what is the rule for g(x)?

Answer 1

My guess is that, if h(x)=2x+4, g(x)=3(2(x+6)+4)=6(x+6)+12=6x+36+12=6x+48.

The effect is to shift the straight line 2x+4 6 units to the left, which means it intersects the x axis at -8 instead of -2; and the line is steeper, so the slope increase from 2 to 6, a factor of 3. This is a stretch in the vertical direction of 3.

Comments

Rudolf decides to make and sell necklaces to earn money to buy a new computer. He plans to charge $4.75 per necklace.

a) Write a function that describes the revenue R(n), in dollars, Rudolf will earn from selling n necklaces.

b) What is a reasonable domain for this function?

c) Graph the function.

d) Identify and interpret the intercepts of the function.

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a) R(n)=4.75n

b) Perhaps Rudolf can make 100 necklaces to buy a computer costing $475, so the domain would be 1 to 100 or perhaps 0 to 100 if he couldn't be bothered to make even one!

c) Straight line passing through the origin with a slope of 4.75.

d) The intercepts are both 0.

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Use the two given functions to choose the best statement comparing the graphs to each other.
Function 1: y=-x-3    Funtion 2: y=x+1
A. Function 2's graph is shifted down 4 units from Function 1's graph and is reflected.
B. Function 2's graph is shifted down 4 units from Function 1's graph and is not reflected.
C. Function 2's graph is shifted up 4 units from Function 1's graph and is reflected.
D. Function 2's graph is shifted up 4 units from Function 1's graph and is not reflected.

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Red line is Function 1, blue line is Function 2.

When x=0 (y axis) the lines are separated by 4 units with 1 sitting below 2, so Function 2 is shifted up from Function 1. The two functions are reflections of one another if by reflection is meant that, for example, (-1,0) is a reflection of (-3,0) and (-1,-2) reflects (-3,-2) where the vertical line x=-2 is a double-sided mirror.

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So, Function 2's graph is shifted up 4 units from Function 1's and is not reflected, correct?

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Not sure about the reflection part because I don't know how reflection is defined.

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Use the two given functions to choose the best statement comparing the graphs to each other.

Function 1: y=-x+3       Function 2: 2x+1

A. Function 2's graph is less steep than Function 1's graph and is reflected.

B. Function 2's graph is steeper than Function 1's graph and is reflected.

C. Function 2's graph is less steep than Function 1's graph and is not reflected.

D. Function 2's graph is steeper than Function 1's graph and is not reflected.

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Your original commented question:

Did some checking. Reflection seems to be defined two ways for functions.

If F1(x)=F2(-x) or F1(x)=-F2(x) then F1 reflects F2 in one or other axis.

Clearly this is not the case for Functions 1 and 2, so you're right, under this definition Function does not reflect Function 2. Everything depends on how reflection is defined.

Now let's see your new question.

I think the answer is D. The coefficient 2 makes Function 2 steeper. By the above definition of reflection, the functions do not reflect each other.