how was the y=x^2 transformed?The blue line is y=x^2the red one y=-28/289(x-21)^2+28 ### Understanding Graph TranslationsIn this lesson, we will explore the concept of translating graphs of functions.#### Example:1. **Equations Displayed:** - $ y = x^2 $ (Blue graph) - $ y = -\frac{28}{289}(x - 21)^2 - 28 $ (Red graph)2. **Graph Descriptions:** - **Blue Graph:** This is a standard parabolic graph representing the quadratic function $ y = x^2 $. The graph opens upwards and is centered at the origin (0,0). - **Red Graph:** This is another parabolic graph represented by the equation $ y = -\frac{28}{289}(x - 21)^2 - 28 $. This graph opens downwards due to the negative leading coefficient and is shifted compared to the blue graph.3. **Key Points:** - The blue graph intersects the y-axis at the origin $(0, 0)$. - The red graph has a vertex at the point $(21, -28)$ due to the translation and transformation of the standard parabola.4. **Translations:** - From $ y = x^2 $ to $ y = -\frac{28}{289}(x - 21)^2 - 28 $: - **Horizontal shift:** The red graph is moved 21 units to the right, evident from the term $ (x - 21) $. - **Vertical shift:** It is moved 28 units down, represented by the -28 outside the squared term. - **Vertical stretch and reflection:** The term $-\frac{28}{289}$ shows that the graph is reflected over the x-axis (negative coefficient) and also stretched vertically by $\frac{28}{289}$.5. **Key Points Identified:** - For the red graph, the points identified are near $ (0, 9) $ and $ (0, 35) $, indicating important intersections or features, which are points where you might need to analyze how the function behaves.#### Instruction:**Find the translations:**Analyze how each term in the equation affects the graph from the original function $ y = x^2 $. Understand how horizontal shifts, vertical shifts, and stretching/compressing change the graph's shape and position.### ConclusionUnderstanding the impact

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Answered: how was the y=x^2 transformed? The blue…

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how was the y=x^2 transformed? The blue line is y=x^2 the red one y=-28/289(x-21)^2+28

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

how was the y=x^2 transformed? The blue line is y=x^2 the red one y=-28/289(x-21)^2+28

### Understanding Graph Translations

In this lesson, we will explore the concept of translating graphs of functions.

#### Example:

1. **Equations Displayed:**
- $ y = x^2 $ (Blue graph)
- $ y = -\frac{28}{289}(x - 21)^2 - 28 $ (Red graph)

2. **Graph Descriptions:**
- **Blue Graph:** This is a standard parabolic graph representing the quadratic function $ y = x^2 $. The graph opens upwards and is centered at the origin (0,0).
- **Red Graph:** This is another parabolic graph represented by the equation $ y = -\frac{28}{289}(x - 21)^2 - 28 $. This graph opens downwards due to the negative leading coefficient and is shifted compared to the blue graph.

3. **Key Points:**
- The blue graph intersects the y-axis at the origin $(0, 0)$.
- The red graph has a vertex at the point $(21, -28)$ due to the translation and transformation of the standard parabola.

4. **Translations:**
- From $ y = x^2 $ to $ y = -\frac{28}{289}(x - 21)^2 - 28 $:
- **Horizontal shift:** The red graph is moved 21 units to the right, evident from the term $ (x - 21) $.
- **Vertical shift:** It is moved 28 units down, represented by the -28 outside the squared term.
- **Vertical stretch and reflection:** The term $-\frac{28}{289}$ shows that the graph is reflected over the x-axis (negative coefficient) and also stretched vertically by $\frac{28}{289}$.

5. **Key Points Identified:**
- For the red graph, the points identified are near $ (0, 9) $ and $ (0, 35) $, indicating important intersections or features, which are points where you might need to analyze how the function behaves.

#### Instruction:

**Find the translations:**

Analyze how each term in the equation affects the graph from the original function $ y = x^2 $. Understand how horizontal shifts, vertical shifts, and stretching/compressing change the graph's shape and position.

### Conclusion

Understanding the impact

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