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.