Explain how the graph of the square root function $$ f(x) = \sqrt{x} $$ shifts when you change it to $$ f(x) = \sqrt{x - 4} $$.

Question

UpStudy AI · Accepted Answer

When you modify the square root function from $$ f(x) = \sqrt{x} $$ to $$ f(x) = \sqrt{x - 4} $$, you're applying a **horizontal shift** to the graph of the function. Here's a detailed explanation of how and why this shift occurs:

### Original Function: $$ f(x) = \sqrt{x} $$
- **Graph Characteristics:**
  - **Starting Point:** The graph begins at the origin $$(0, 0)$$, since $$ \sqrt{0} = 0 $$.
  - **Domain:** $$ x \geq 0 $$ (you can't take the square root of a negative number in the set of real numbers).
  - **Shape:** The graph increases gradually, creating the familiar "half-parabola" shape on its side.

### Modified Function: $$ f(x) = \sqrt{x - 4} $$
- **Transformation Applied:** $$ x $$ is replaced with $$ x - 4 $$ inside the square root.
- **Effect of the Transformation:**
  - **Horizontal Shift:** Replacing $$ x $$ with $$ x - 4 $$ shifts the entire graph **4 units to the right**. This is because for any given output value of the function, the input $$ x $$ must now be 4 units larger to achieve the same result.
  
    - **General Rule:** In the function $$ f(x - h) $$, the graph of $$ f(x) $$ shifts **h units to the right**.
  
- **New Starting Point:** The graph now begins at $$ (4, 0) $$, since:
  $$
  \sqrt{4 - 4} = \sqrt{0} = 0
  $$
  
- **New Domain:** $$ x - 4 \geq 0 $$ implies $$ x \geq 4 $$.

### Visualizing the Shift
Consider some specific points on the original graph and see how they move:

| Original Function $$ f(x) = \sqrt{x} $$ | Shifted Function $$ f(x) = \sqrt{x - 4} $$ |
|------------------------------------------|----------------------------------------------|
| $$ (0, 0) $$                             | $$ (4, 0) $$                                 |
| $$ (1, 1) $$                             | $$ (5, 1) $$                                 |
| $$ (4, 2) $$                             | $$ (8, 2) $$                                 |

Each point $$(a, b)$$ on the original graph moves to $$(a + 4, b)$$ on the shifted graph.

### Summary of the Shift
- **Direction:** Right
- **Distance:** 4 units
- **Reason:** The expression $$ x - 4 $$ inside the square root indicates a horizontal translation according to the transformation rules for functions.

### Graphical Representation

Below is a simple schematic to illustrate the shift:

```
Original $$ f(x) = \sqrt{x} $$:
|
|        *
|      *
|    *
|  *
|*_________________
(0,0)

Shifted $$ f(x) = \sqrt{x - 4} $$:
|
|              *
|            *
|          *
|        *
|      *
|_____________________
           (4,0)
```

In the schematic:
- The asterisks (*) represent points on the graph.
- The shifted graph starts at $$ (4, 0) $$ instead of $$ (0, 0) $$, illustrating the rightward movement.

### Conclusion

Changing the function from $$ f(x) = \sqrt{x} $$ to $$ f(x) = \sqrt{x - 4} $$ results in shifting the entire graph **4 units to the right**. This horizontal translation adjusts the starting point and domain of the function accordingly.
The graph of $$ f(x) = \sqrt{x} $$ shifts 4 units to the right when changed to $$ f(x) = \sqrt{x - 4} $$.

Explain how the graph of the square root function \( f(x) = \sqrt{x} \) shifts when you change it to \( f(x) = \sqrt{x - 4} \).

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions