Use transformations of $$ f(x)=x^{2} $$ to graph the following function. $$ g(x)=x^{2}-5 $$

Question

UpStudy AI · Accepted Answer

To graph the function $$ g(x) = x^{2} - 5 $$ using transformations of the parent function $$ f(x) = x^{2} $$, follow these steps:

### 1. Understand the Parent Function
The parent function $$ f(x) = x^{2} $$ is a basic parabola with:
- **Vertex** at the origin $$(0, 0)$$.
- **Axis of symmetry** along the y-axis.
- **Opens upward**, meaning it curves upwards on both sides of the vertex.

### 2. Identify the Transformation
The function $$ g(x) = x^{2} - 5 $$ involves a **vertical shift** of the parent function. Specifically:
- The "-5" indicates that the entire graph of $$ f(x) = x^{2} $$ is shifted **downward** by 5 units.

### 3. Apply the Transformation
- **Vertex Shift**: The original vertex $$(0, 0)$$ moves down 5 units to $$(0, -5)$$.
  
  $$
  \text{Original Vertex: } (0, 0) \quad \rightarrow \quad \text{New Vertex: } (0, 0 - 5) = (0, -5)
  $$

- **Shape and Orientation**: The shape of the parabola remains unchanged, and it still **opens upward**.

### 4. Plot Key Points
To accurately graph $$ g(x) = x^{2} - 5 $$, plot several points:

| $$x$$ | $$ f(x) = x^{2} $$ | $$ g(x) = x^{2} - 5 $$ |
|-------|--------------------|------------------------|
| -2    | 4                  | $$ 4 - 5 = -1 $$       |
| -1    | 1                  | $$ 1 - 5 = -4 $$       |
| 0     | 0                  | $$ 0 - 5 = -5 $$       |
| 1     | 1                  | $$ 1 - 5 = -4 $$       |
| 2     | 4                  | $$ 4 - 5 = -1 $$       |

**Plot these points:**
- $$(-2, -1)$$
- $$(-1, -4)$$
- $$(0, -5)$$
- $$(1, -4)$$
- $$(2, -1)$$

### 5. Draw the Graph
1. **Plot the new vertex** at $$(0, -5)$$.
2. **Plot the additional points** from the table above.
3. **Draw a symmetric parabola** through these points, maintaining the upward orientation.

### 6. Visual Representation
While I can't provide a visual graph here, here's a description to help you sketch it:

- Start by drawing the standard parabola $$ f(x) = x^{2} $$ with its vertex at the origin.
- Shift the entire graph **downward** by 5 units so that the vertex is now at $$(0, -5)$$.
- Ensure that the shape remains the same, with the parabola still opening upward.
- Plot the additional points to confirm the accuracy of the transformation.

### 7. Summary of the Transformation
- **Type of Transformation**: Vertical Shift
- **Direction**: Downward
- **Magnitude**: 5 units
- **New Vertex**: $$(0, -5)$$

### 8. Final Graph Features
- **Vertex**: $$(0, -5)$$
- **Axis of Symmetry**: $$x = 0$$ (y-axis)
- **Opens**: Upward
- **Y-intercept**: $$(0, -5)$$ (same as vertex)
- **X-intercepts**: Solve $$ x^{2} - 5 = 0 $$:
  
  $$
  x^{2} = 5 \quad \Rightarrow \quad x = \pm \sqrt{5} \approx \pm 2.24
  $$
  
  So, the x-intercepts are approximately $$(2.24, 0)$$ and $$(-2.24, 0)$$.

By following these steps, you can accurately graph $$ g(x) = x^{2} - 5 $$ using the transformations of the parent function $$ f(x) = x^{2} $$.
To graph $$ g(x) = x^{2} - 5 $$, shift the parent function $$ f(x) = x^{2} $$ downward by 5 units. This moves the vertex from $$(0, 0)$$ to $$(0, -5)$$ and keeps the parabola's shape and upward orientation unchanged. Plot key points and draw the symmetric parabola accordingly.

Use transformations of \( f(x)=x^{2} \) to graph the following function. \[ g(x)=x^{2}-5 \]

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Algebra Questions