The given quadratic function is of the form:

$f(x) = a(x - h)^2 + k$

We need to determine whether each constant $a, h,$ and $k$ is positive, negative, or zero by analyzing the graph.

Analysis:

1. Determining $a$:

   • The value of $a$ controls the direction of the parabola's opening. If $a > 0$, the parabola opens upwards. If $a < 0$, it opens downwards.
   • In the graph provided, the parabola opens downwards, indicating that $a$ is negative.

2. Determining $h$:

   • The value of $h$ is the x-coordinate of the vertex of the parabola.
   • From the graph, the vertex is on the right side of the y-axis, meaning $h > 0$. Hence, $h$ is positive.

3. Determining $k$:

   • The value of $k$ is the y-coordinate of the vertex.
   • The vertex is below the x-axis, indicating $k < 0$. Thus, $k$ is negative.

Final Answers:

• $a$ is negative.
• $h$ is positive.
• $k$ is negative.

Let me know if you'd like more details or have any questions!

Related Questions:

1. How does changing $a$ affect the shape and width of the parabola?
2. What is the effect of $h$ on the horizontal position of the vertex?
3. How does $k$ influence the vertical translation of the graph?
4. What happens if $a = 0$? How would the graph look?
5. How can we find the axis of symmetry using $h$?

Tip: For any quadratic function in vertex form, the vertex is located at $(h, k)$.