It looks like you are working on a worksheet involving the graphing and transformations of absolute value equations. Let's break down the questions and help solve them step by step.

1. Matching absolute value functions with graphs (Questions 1–6):

You are given three absolute value functions and several graphs. To match these, we need to analyze the transformations (shifts and reflections) in each equation.

• $f(x) = -|x - 2|$:

  • This has a horizontal shift of 2 units to the right (because of $x - 2$) and a reflection over the x-axis (because of the negative sign).

• $f(x) = |x - 2| + 2$:

  • This has a horizontal shift of 2 units to the right and a vertical shift of 2 units up.

• $f(x) = -|x + 2| - 2$:

  • This has a horizontal shift of 2 units to the left (because of $x + 2$) and a reflection over the x-axis. There's also a vertical shift of 2 units down.

Once you analyze these transformations, you can match the graphs accordingly.

2. Problem 7: $f(x) = |x - 5| - 3$

Now, let's break down this function and fill out the required information.

• A. Graph opens:

  • Since the absolute value function is positive (no negative sign in front), the graph opens up.

• B. Vertex:

  • The vertex comes from the transformation of the basic absolute value function. The function $f(x) = |x - 5| - 3$ is shifted 5 units to the right and 3 units down, so the vertex is (5, -3).

• C. Axis of symmetry:

  • The axis of symmetry is the vertical line passing through the vertex, so the axis of symmetry is x = 5.

• D. Slope:

  • The slope of the arms of the "V" is 1 for the right arm and -1 for the left arm (as this is a basic absolute value function without any stretching).

• E. Domain:

  • The domain of any absolute value function is all real numbers, so the domain is (-∞, ∞).

• F. Range:

  • Since the graph opens upward and the vertex is at y = -3, the range is [-3, ∞).

Table of values and graph:

You can plot points by choosing a few values of $x$, plugging them into the equation $f(x) = |x - 5| - 3$, and finding the corresponding $y$-values. Here's a sample table:

| $x$ | $y = |x - 5| - 3$ | |----------|------------------------| | 3 | -1 | | 4 | -2 | | 5 | -3 | | 6 | -2 | | 7 | -1 |

You can then plot these points on the coordinate plane and draw the V-shaped graph with a vertex at (5, -3).

Would you like more detailed help with graphing or solving any of these questions?

5 Related Questions:

1. How does the graph of $f(x) = |x|$ change when transformed into $f(x) = |x - h| + k$?
2. What effect does the negative sign in front of an absolute value function, like $-|x|$, have on the graph?
3. Can you describe how to determine the vertex and axis of symmetry for any absolute value function?
4. How do you compute the slope of the arms in an absolute value function?
5. Why does the domain of absolute value functions always cover all real numbers?

Tip:

Remember, the vertex form of an absolute value function $f(x) = a|x - h| + k$ makes it easy to spot transformations like shifts and reflections.