How do you graph #y=abs(x+2)-3#?

Question

Nathan Axel · Accepted Answer

see explanation

graph{y=|x+2|-3 [-10, 10, -5, 5]} To find the points on a graph, plug in the #x# values to get the matching #y# values. For example: #y=|(1)+2|-3# #y=|3|-3# #y=0# point: #(1,0)# #y=|(-4)+2|-3# #y=|-2|-3# #y=-1# point: #(-4,-1)#

Savannah Anderson · Answer

undefined To graph $y = |x + 2| - 3$, follow these steps:

1. **Understand the Function**: The absolute value function $|x + 2|$ shifts the basic $|x|$ graph 2 units to the left. Subtracting 3 from $|x + 2|$ then shifts the graph down by 3 units.

2. **Plot the Vertex**: The vertex of the absolute value function $|x + 2|$ is at $(-2, 0)$. After applying the -3 shift, the vertex moves to $(-2, -3)$. Plot this point on your graph.

3. **Create a Table of Values**: Choose values for $x$ to the left and right of -2, and compute the corresponding $y$ values. For example:
   - For $x = -4$: $y = |(-4) + 2| - 3 = |-2| - 3 = 2 - 3 = -1$.
   - For $x = 0$: $y = |0 + 2| - 3 = |2| - 3 = 2 - 3 = -1$.

4. **Plot Additional Points**: Plot the points you calculated in the step above. For $x = -4$, plot (-4, -1). For $x = 0$, plot (0, -1).

5. **Draw the Graph**: Connect the points with two straight lines that meet at the vertex $(-2, -3)$. The lines should form a "V" shape, opening upwards because the absolute value function produces non-negative outputs, which are then adjusted downwards by 3 units.

6. **Check for Symmetry**: The graph of an absolute value function is always symmetric with respect to the vertical line through its vertex. In this case, that's the line $x = -2$.

By following these steps, you'll have the graph of $y = |x + 2| - 3$, with a vertex at $(-2, -3)$ and lines extending upward and outward from the vertex, creating a "V" shape.