The equation given is:

$f(x) = -2|x - 4| + 2$

This is an absolute value function of the form $f(x) = a|x - h| + k$, where:

• $a = -2$, which controls the vertical stretch or compression and the reflection over the x-axis.
• $h = 4$, which is the horizontal shift, moving the vertex to $x = 4$.
• $k = 2$, which is the vertical shift, moving the vertex up by 2 units.

Key points to graph this function:

1. Vertex: The vertex is located at $(h, k) = (4, 2)$.
2. Slope: Since $a = -2$, the slope of the lines is $-2$ on both sides of the vertex. On the left side (for $x < 4$), the function will have a positive slope (rising), and on the right side (for $x > 4$), it will have a negative slope (falling).
3. Reflection: The negative sign in front of the 2 means the graph will open downward.

Steps for graphing:

1. Plot the vertex at (4, 2).

2. Plot additional points:

   • For $x = 3$: $f(3) = -2|3 - 4| + 2 = -2(1) + 2 = 0$. So the point is $(3, 0)$.
   • For $x = 5$: $f(5) = -2|5 - 4| + 2 = -2(1) + 2 = 0$. So the point is $(5, 0)$.
   • For $x = 2$: $f(2) = -2|2 - 4| + 2 = -2(2) + 2 = -4 + 2 = -2$. So the point is $(2, -2)$.
   • For $x = 6$: $f(6) = -2|6 - 4| + 2 = -2(2) + 2 = -4 + 2 = -2$. So the point is $(6, -2)$.

3. Draw the graph: Connect the points with two straight lines that form a "V" shape, opening downward, centered at the vertex $(4, 2)$.

Would you like more details on this graph, or do you have any questions?

Related questions:

1. How does changing the value of $a$ affect the shape of the absolute value graph?
2. What happens if we change the value of $h$ in the equation $f(x) = -2|x - h| + 2$?
3. How would the graph change if the constant $k$ were increased or decreased?
4. How can we find the x-intercepts of the graph?
5. What does the reflection of the graph tell us about the function's behavior?

Tip:

When graphing absolute value functions, always locate the vertex first, and then use the slope to plot points symmetrically on both sides of the vertex.