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The given problem requires sketching the graphs of three functions on the same coordinate plane and verifying with a graphing utility. The functions provided are:

1. \( f(x) = (x - 3)^2 \)
2. \( g(x) = (x + 3)^2 + 3 \)
3. \( h(x) = -(x - 3)^2 - 4 \)

Let's break down each function:

### 1. \( f(x) = (x - 3)^2 \)
This is a standard quadratic function in vertex form, shifted to the right by 3 units. The vertex is at \( (3, 0) \), and since the coefficient of \( (x - 3)^2 \) is positive, the parabola opens upwards.

### 2. \( g(x) = (x + 3)^2 + 3 \)
This is also a quadratic function in vertex form, shifted to the left by 3 units (due to \( x + 3 \)) and up by 3 units (because of the \( +3 \) outside the square). The vertex is at \( (-3, 3) \), and the parabola opens upwards.

### 3. \( h(x) = -(x - 3)^2 - 4 \)
This quadratic function is reflected over the x-axis (because of the negative sign in front of the square), and it is shifted 3 units to the right and 4 units downward. The vertex is at \( (3, -4) \), and the parabola opens downward.

### Summary of Transformations:
- \( f(x) = (x - 3)^2 \): Shifted right 3 units, vertex at \( (3, 0) \), opens upwards.
- \( g(x) = (x + 3)^2 + 3 \): Shifted left 3 units and up 3 units, vertex at \( (-3, 3) \), opens upwards.
- \( h(x) = -(x - 3)^2 - 4 \): Shifted right 3 units and down 4 units, vertex at \( (3, -4) \), opens downwards.

### Sketching:
1. For \( f(x) \), plot the vertex at \( (3, 0) \) and sketch an upward-opening parabola.
2. For \( g(x) \), plot the vertex at \( (-3, 3) \) and sketch another upward-opening parabola.
3. For \( h(x) \), plot the vertex at \( (3, -4) \) and sketch a downward-opening parabola.

Would you like me to generate the graphs or explain the next steps in more detail?

### Related Questions:
1. How do you find the vertex of a quadratic function in vertex form?
2. What effect does the sign of the leading coefficient have on the direction of a parabola?
3. How do horizontal and vertical shifts affect the graph of a quadratic function?
4. Can the vertex of a quadratic function be used to solve maximum or minimum problems?
5. How do reflections over the x-axis affect the shape of a parabola?

### Tip:
When sketching quadratic functions, always identify the vertex and axis of symmetry first, as they give the key points to guide the sketch.

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.

Learn how to sketch graphs of three quadratic functions with different transformations, including horizontal and vertical shifts and reflections. This problem is suitable for high school students in Grades 9-12.

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