Question

question 2
0/1 pt 3 2 4 details
function transformations
which of the following functions match the graph to the right?

• ( f(x) = |x - 1| + 3 )
• ( f(x) = |x + 1| + 3 )
• ( f(x) = |x - 3| - 1 )
• ( f(x) = |x + 3| - 1 )

question help: message instructor
add work
submit question

Explanation:

Step1: Recall vertex form of absolute value function

The general form of an absolute value function is \( f(x) = |x - h| + k \), where \((h, k)\) is the vertex of the V - shaped graph.

Step2: Identify the vertex from the graph

Looking at the graph, the vertex (the point of the V) is at \((- 3,-1)\)? Wait, no, wait. Wait, let's check the graph again. Wait, the x - axis: the grid lines. Wait, the vertex seems to be at (x=-3? No, wait, looking at the graph, the vertex is at (x = - 3? Wait, no, let's see the x - coordinates. Wait, the graph's vertex: let's check the options. Wait, the vertex of \( f(x)=|x - h|+k \) is (h,k). Let's check each option:

• For \( f(x)=|x - 1|+3 \), vertex is (1,3)
• For \( f(x)=|x + 1|+3 \), vertex is (- 1,3)
• For \( f(x)=|x - 3|-1 \), vertex is (3,-1)
• For \( f(x)=|x + 3|-1 \), vertex is (-3,-1)

Wait, maybe I misread the graph. Wait, the user's graph: let's see the x - axis. The graph has the vertex at x=-3? No, wait, maybe I made a mistake. Wait, no, let's check the graph again. Wait, the graph is a V - shape. Let's look at the options. Wait, the correct vertex: let's see the y - coordinate. The vertex is at y=-1? Wait, no, maybe the vertex is at (x=-3, y=-1)? Wait, no, let's check the options. Wait, the function \( f(x)=|x + 3|-1 \) has vertex at (-3,-1). Let's confirm:

The absolute value function \( y = |x - h|+k \) has vertex (h,k). So \( y=|x + 3|-1=|x-(-3)|+(-1) \), so vertex is (-3,-1). Let's check the graph: if the vertex is at (-3,-1), then that's the point of the V. So the correct function should be \( f(x)=|x + 3|-1 \).

Wait, maybe I misread the graph earlier. Let's re - evaluate. Let's check each option's vertex:

1. \( f(x)=|x - 1|+3 \): vertex (1,3)
2. \( f(x)=|x + 1|+3 \): vertex (-1,3)
3. \( f(x)=|x - 3|-1 \): vertex (3,-1)
4. \( f(x)=|x + 3|-1 \): vertex (-3,-1)

Now, looking at the graph, the vertex is at (x=-3, y=-1), so the function with vertex (-3,-1) is \( f(x)=|x + 3|-1 \).

Answer:

\( f(x)=|x + 3|-1 \) (the fourth option: \( f(x)=\boldsymbol{|x + 3|-1} \))