Question

the graph below shows a vertical stretch of \\(\square\\) factor performed on the function \\(y = |x - 1| - 4\\). use an integer in your answer.

Explanation:

Step1: Recall vertex form of absolute value function

The general form of a vertically stretched absolute value function is $y = a|x - h| + k$, where $a$ is the vertical stretch factor, $(h,k)$ is the vertex.

Step2: Identify vertex of given function

For $y = |x - 1| - 4$, the vertex is $(1, -4)$.

Step3: Find vertex from the graph

From the graph, the vertex is $(1, -4)$ (matches the base function's vertex), and another point on the graph is $(2, -1)$.

Step4: Substitute point into stretched function

Substitute $x=2$, $y=-1$, $h=1$, $k=-4$ into $y = a|x - 1| - 4$:
$-1 = a|2 - 1| - 4$

Step5: Solve for stretch factor $a$

Simplify and solve:
$-1 = a(1) - 4$
$a = -1 + 4 = 3$

Answer:

3