Question

Question 1

Create three graphs. The first should show the parent function for linear equations and the second should show the parent function for a quadratic function. For the third graph, you can create either a square root, cube root, or exponential function.

Create three options for each of the graphs
Question 2

Using your first graph (the linear parent function), create a table. In the table, show what happens to the values when you change the function to y=−x
. Graph the new function. In 1–2 sentences, describe the transformation that occurred.

Question 3

Use your quadratic graph to set up four tables. Find the values for the transformations y=f(x)+k
, y=f(x)−k
, y=kf(x)
, and y=f(kx)
. You can choose any value for k but use the same value for each transformation. Use your tables to graph each new function. In 1–2 sentences each, describe what happens to the parent function in each transformation.

Question 4

Using your final parent graph, conduct at least three different transformations. Use a table, a graph, and a written explanation to describe each transformation.

Answer 1

### Question 1

#### Graph 1: Linear Parent Function

Option 1: Linear Function
\( y = x \)

Option 2: Linear Function
\( y = 2x \)

Option 3: Linear Function
\( y = -0.5x \)


#### Graph 2: Quadratic Parent Function

Option 1: Quadratic Function
\( y = x^2 \)

Option 2: Quadratic Function
\( y = (x - 1)^2 \)
^2)

Option 3: Quadratic Function
\( y = -x^2 + 3 \)


#### Graph 3: Exponential Function

Option 1: Exponential Function
\( y = 2^x \)

Option 2: Exponential Function
\( y = 3^x \)

Option 3: Exponential Function
\( y = e^x \)

---

### Question 2

#### Table for the Linear Function \( y = x \) vs. \( y = -x \)

| x | \( y = x \) | \( y = -x \) |
|-----|-------------|---------------|
| -2 | -2 | 2 |
| -1 | -1 | 1 |
| 0 | 0 | 0 |
| 1 | 1 | -1 |
| 2 | 2 | -2 |

#### Explanation of Transformation

When transforming the function from \( y = x \) to \( y = -x \), the graph reflects across the y-axis. The output values are the negatives of the original input values.

---

### Question 3

#### Quadratic Function Transformations

##### Table for \( y = f(x) + k \)

Let's choose \( k = 2 \).

| x | \( y = x^2 \) | \( y = x^2 + 2 \) |
|-----|---------------|-------------------|
| -2 | 4 | 6 |
| -1 | 1 | 3 |
| 0 | 0 | 2 |
| 1 | 1 | 3 |
| 2 | 4 | 6 |

##### Explanation:

The graph shifts up by 2 units.

---

##### Table for \( y = f(x) - k \)

| x | \( y = x^2 \) | \( y = x^2 - 2 \) |
|-----|---------------|-------------------|
| -2 | 4 | 2 |
| -1 | 1 | -1 |
| 0 | 0 | -2 |
| 1 | 1 | -1 |
| 2 | 4 | 2 |

##### Explanation:

The graph shifts down by 2 units.

---

##### Table for \( y = kf(x) \)

| x | \( y = x^2 \) | \( y = 3x^2 \) |
|-----|---------------|-----------------|
| -2 | 4 | 12 |
| -1 | 1 | 3 |
| 0 | 0 | 0 |
| 1 | 1 | 3 |
| 2 | 4 | 12 |

##### Explanation:

The graph vertically stretches by a factor of 3.

---

##### Table for \( y = f(kx) \)

| x | \( y = x^2 \) | \( y = (2x)^2 \) |
|-----|---------------|-------------------|
| -2 | 4 | 16 |
| -1 | 1 | 4 |
| 0 | 0 | 0 |
| 1 | 1 | 4 |
| 2 | 4 | 16 |

##### Explanation:

The graph horizontally shrinks by a factor of 2.

---

### Question 4

#### Final Transformations on the Quadratic Graph

1. **Transformation: Vertical Shift up by 2**
- **Table:**

| x | \( y = x^2 \) | \( y = x^2 + 2 \) |
|-----|---------------|-------------------|
| -2 | 4 | 6 |
| -1 | 1 | 3 |
| 0 | 0 | 2 |
| 1 | 1 | 3 |
| 2 | 4 | 6 |

- **Graph:**

- **Explanation:** The graph is shifted 2 units up.

---

2. **Transformation: Vertical Stretch by 2**
- **Table:**

| x | \( y = x^2 \) | \( y = 2x^2 \) |
|-----|---------------|-----------------|
| -2 | 4 | 8 |
| -1 | 1 | 2 |
| 0 | 0 | 0 |
| 1 | 1 | 2 |
| 2 | 4 | 8 |

- **Graph:**

- **Explanation:** The graph stretches vertically, becoming narrower.

---

3. **Transformation: Horizontal Shift left by 1**
- **Table:**

| x | \( y = x^2 \) | \( y = (x + 1)^2 \) |
|-----|---------------|---------------------|
| -3 | 9 | 4 |
| -2 | 4 | 1 |
| -1 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 1 | 4 |

- **Graph:**
^2)

- **Explanation:** The graph shifts 1 unit to the left.