How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Question

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question.

It's telling us to explain the changes that were made to the first function $y = |x + 3| + 4$ to get the second function $y = {1\over4}|2x + 3| + 4$.

What my friend said was that $y = {1\over4}|2x + 3| + 4 \Rightarrow y = {1\over4}|2(x + {3\over2})| + 4$ so there was a vertical stretch of ${1\over4}$, a horizontal stretch of ${1\over2}$, and a horizontal shift of ${3\over2}$ units right.

However, I had a slightly different answer.

I set the first equation as $f(x)$ instead of $y$. So $f(x) = |x + 3| + 4$.

Then:

$f(2x) = |2x + 3| + 4$

$\Rightarrow {1\over4}f(2x) = {1\over4}|2x + 3| + 1$

$\Rightarrow {1\over4}f(2x) + 3 = {1\over4}|2x + 3| + 4$

This means that ${1\over4}f(2x) + 3$ is exactly same as the second equation $y = {1\over4}|2x + 3| + 4$. This implies that the first graph was vertically stretched by ${1\over4}$, horizontally stretched by ${1\over2}$ and vertically shifted up by 3 units.

Since we did not get the same answer, I'm assuming at least one of us are wrong. I'm pretty confident I'm right but so is my friend. I just wanted to see who made the error assuming one of us are right.

Thank you in advance.

Answer 1

You are right and your friend is wrong. The following reasoning will help explain why in a more concrete fashion.

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Original function: $f(x)=|x+3|+4$

New function: $f(x) = \frac{1}{4}|2x+3|+4$

As a simple test, consider what happens when $x=-4$:

Original function at $x=-4$: $$ f(-4)=|-4+3|+4=5 $$ Thus, for the original function, you have the ordered pair $(-4,5)$.

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What you and your friend say:

• (1) Original graph was vertically stretched by $\frac{1}{4}$.
• (2) Original graph was horizontally stretched by $\frac{1}{2}$.
• (3a) Original graph was shifted to the right by $\frac{3}{2}$ units.
• (3b) Original graph was shifted up by $3$ units.

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What actually happens and why you are right and your friend is wrong: Your friend's reasoning gives the following: $$ (-4,5)\overset{\text{(1)}}{\Longrightarrow} (-4,5/4)\overset{\text{(2)}}{\Longrightarrow} (-2,5/4)\overset{\text{(3a)}}{\Longrightarrow} (-7/2,5/4). $$
Your reasoning gives the following: $$ (-4,5)\overset{\text{(1)}}{\Longrightarrow} (-4,5/4)\overset{\text{(2)}}{\Longrightarrow} (-2,5/4)\overset{\text{(3b)}}{\Longrightarrow} (-2,17/4) $$ What does this mean? It means the following in terms of the new function $f(x)=\frac{1}{4}|2x+3|+4$:

• Your friend: $f(-7/2) = 5/4$
• You: $f(-2)=17/4$

Checking the actual calculations confirms your reasoning. We have that $f(-2)=17/4$ while $f(-7/2) = 5 \neq 5/4$.

Answer 2

$$|x+3| + 4 = 1\cdot |x+3| + 4= \frac{2}{2}\cdot |x+3| + 4= \frac{1}{2}|2x+6| + 4 \neq \frac12|2x+3| + 4$$