The absolute value function f is defined by an equation in the form \(\mathrm{f(x) = a|x + b| + c}\),...

1. TRANSLATE the graph information

Looking at the graph carefully:

• The vertex (the lowest point of the V-shape) is at \(\mathrm{(-2, 1)}\)
• The function also passes through \(\mathrm{(-4, 3)}\) on the left side
• The function also passes through \(\mathrm{(0, 3)}\) on the right side

These coordinates will help us determine all the parameters.

2. INFER which parameters we can find from the vertex

Since the general form is \(\mathrm{f(x) = a|x + b| + c}\), and we know the vertex formula:

• The vertex of \(\mathrm{f(x) = a|x + b| + c}\) is at the point \(\mathrm{(-b, c)}\)
• Our vertex is at \(\mathrm{(-2, 1)}\)
• This tells us immediately:
  • \(\mathrm{-b = -2}\), so \(\mathrm{b = 2}\)
  • \(\mathrm{c = 1}\), so \(\mathrm{c = 1}\)

So far we have: \(\mathrm{f(x) = a|x + 2| + 1}\)

3. INFER how to find the missing parameter a

We need one more piece of information to find \(\mathrm{a}\):

• The vertex alone doesn't tell us the steepness (the value of \(\mathrm{a}\))
• We need to use another point from the graph
• Let's use the point \(\mathrm{(-4, 3)}\)

4. SIMPLIFY to find a

Substitute the point \(\mathrm{(-4, 3)}\) into \(\mathrm{f(x) = a|x + 2| + 1}\):

• \(\mathrm{f(-4) = 3}\)
• \(\mathrm{a|-4 + 2| + 1 = 3}\)
• \(\mathrm{a|-2| + 1 = 3}\)
• \(\mathrm{a(2) + 1 = 3}\)
• \(\mathrm{2a = 2}\)
• \(\mathrm{a = 1}\)

Now we have the complete function: \(\mathrm{f(x) = |x + 2| + 1}\)

5. SIMPLIFY the transformation to find g(x)

We're told that \(\mathrm{g(x) = f(x - 3)}\). This means we replace every \(\mathrm{x}\) in \(\mathrm{f(x)}\) with \(\mathrm{(x - 3)}\):

• \(\mathrm{g(x) = f(x - 3) = |(x - 3) + 2| + 1}\)
• SIMPLIFY inside the absolute value:
  • \(\mathrm{(x - 3) + 2 = x - 1}\)
• So: \(\mathrm{g(x) = |x - 1| + 1}\)

6. TRANSLATE g(x) into the required form

We need to express \(\mathrm{g(x)}\) in the form \(\mathrm{a|x + b| + c}\):

• \(\mathrm{g(x) = |x - 1| + 1}\)
• Rewrite \(\mathrm{x - 1}\) as \(\mathrm{x + (-1)}\):
  • \(\mathrm{g(x) = 1|x + (-1)| + 1}\)
• Therefore, for function \(\mathrm{g}\):
  • \(\mathrm{a = 1}\)
  • \(\mathrm{b = -1}\)
  • \(\mathrm{c = 1}\)

7. Calculate the final answer

• \(\mathrm{a + b + c = 1 + (-1) + 1 = 1}\)

Answer: 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: After finding \(\mathrm{g(x) = |x - 1| + 1}\), students incorrectly identify \(\mathrm{b = 1}\) instead of \(\mathrm{b = -1}\)

Students see \(\mathrm{|x - 1|}\) and think "the number next to x is -1, but b must be positive, so b = 1." However, they need to rewrite this in the form \(\mathrm{|x + b|}\):

• \(\mathrm{|x - 1| = |x + (-1)|}\)
• Therefore \(\mathrm{b = -1}\), not \(\mathrm{b = 1}\)

With the incorrect \(\mathrm{b = 1}\), they would calculate: \(\mathrm{a + b + c = 1 + 1 + 1 = 3}\)

This leads to an incorrect answer of 3.

Second Most Common Error:

Weak INFER skill combined with poor SIMPLIFY execution: Students confuse which parameters belong to \(\mathrm{f(x)}\) versus \(\mathrm{g(x)}\), and calculate \(\mathrm{a + b + c}\) using the original function's parameters

They correctly find that \(\mathrm{f(x) = |x + 2| + 1}\), so for \(\mathrm{f}\): \(\mathrm{a = 1, b = 2, c = 1}\)

But then they forget to apply the transformation and simply calculate: \(\mathrm{a + b + c = 1 + 2 + 1 = 4}\)

This leads to an incorrect answer of 4, using the parameters from the wrong function.

The Bottom Line:

This problem requires careful tracking of two different functions (\(\mathrm{f}\) and \(\mathrm{g}\)) and precise algebraic manipulation when rewriting expressions in standard form. The transformation \(\mathrm{f(x - 3)}\) shifts the graph but students must recognize that this changes the internal expression, affecting the value of \(\mathrm{b}\) while \(\mathrm{a}\) and \(\mathrm{c}\) remain the same in magnitude. The key pitfall is the sign: \(\mathrm{|x - 1|}\) must be rewritten as \(\mathrm{|x + (-1)|}\) to correctly identify \(\mathrm{b = -1}\).