Given the function f(x) = (x - 4)2 - 3

Check all that apply


this is an exponential function

there is a shift left 4 units

there is a shift right 4 units

there is a reflection across x-axis

there is a shift down 3 units

there is a shift up 3 units

the domain is (-inf,inf)

the range is (-inf,-3]

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!

Given the function f(x) = (x - 4)^2 - 3, check all that apply: this is an exponential function, there is a shift left 4 units, there is a shift right 4 units, there is a reflection across the x-axis, there is a shift down 3 units, there is a shift up 3 units, the domain is (-inf,inf), the range is (-inf,-3].

Explore the properties of the quadratic function f(x) = (x - 4)^2 - 3. Understand key concepts such as graph shifts, domain, and range. This problem is ideal for students in Grades 9-10 learning about quadratic functions and transformations.

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