Given the function $f(x) = (x - 4)^2 - 3$, let's analyze the characteristics:

1. This is an exponential function:

   • Incorrect. The given function is a quadratic function (of the form $(x - h)^2 + k$), not exponential. Quadratic functions have $x^2$ terms, while exponential functions involve expressions like $a^x$.

2. There is a shift left 4 units:

   • Incorrect. The term $(x - 4)$ indicates a shift right 4 units (not left). Shifts to the right occur when subtracting a positive value from $x$.

3. There is a shift right 4 units:

   • Correct. The term $(x - 4)$ means the graph of $x^2$ is shifted 4 units to the right.

4. There is a reflection across the x-axis:

   • Incorrect. There is no negative sign in front of the squared term, so no reflection occurs across the x-axis.

5. There is a shift down 3 units:

   • Correct. The $-3$ at the end of the function shifts the graph 3 units downward.

6. There is a shift up 3 units:

   • Incorrect. The function has a shift downward, not upward.

7. The domain is $(-\infty, \infty)$:

   • Correct. Quadratic functions are defined for all real values of $x$, so the domain is $(-\infty, \infty)$.

8. The range is $(-\infty, -3]$:

   • Incorrect. The parabola opens upwards, and its minimum value is at $-3$, so the range is $[-3, \infty)$, not $(-\infty, -3]$.

Summary of correct answers:

• There is a shift right 4 units.
• There is a shift down 3 units.
• The domain is $(-\infty, \infty)$.

Would you like more details on any of these?

Here are 5 related questions to further explore these ideas:

1. How can we determine the vertex of a quadratic function?
2. What distinguishes a quadratic function from an exponential function?
3. How do we find the axis of symmetry for a parabola?
4. What are the effects of reflections in quadratic functions?
5. How does a function's range change when its graph is shifted?

Tip: In any quadratic function, the vertex form $(x - h)^2 + k$ gives immediate insight into its shifts and vertex location!