Question: Directions: Read the situation carefully. Answer each guide question completely. Show your solutions whenever necessary. Write your answers in the spaces provided. Situation: A civil engineer is designing a pedestrian pathway in a community park. The original pathway is represented by the equation. To improve the park layout, the engineer applies different graph transformations to the pathway. Guide Questions 1. The engineer translates the pathway 5 units upward. Write the new equation. 2. A trainee says, "The translated pathway has the same slope as the original pathway." Do you agree? Explain your answer. 3. The engineer translates the original pathway 4 units downward. Write the new equation. 4. Compare your answers in Items 1 and 3. What part of the equation changed? Explain. 5. The engineer reflects the original pathway across the x-axis. Write the new equation. 6. Another trainee says, "Reflecting a graph across the x-axis changes only the sign of the y-values." Do you agree? Explain your answer. 7. The engineer reflects the original pathway across the y-axis. Write the new equation. 8. Compare the equations obtained in Items 5 and 7. How are the two reflected equations different? 9. Which transformation performed in this activity changes only the y-intercept while keeping the slope the same? Explain your answer. 10. Suppose another engineer wants the new pathway to remain parallel to the original pathway. Which transformation(s) from this activity would satisfy this condition? Explain why.

1. **State the problem:** The original pathway is represented by an equation of a line, which we assume to be $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. 2. **Translate 5 units upward:** Translating a graph upward by 5 units means adding 5 to the y-values. New equation: $$y = mx + b + 5$$ 3. **Slope comparison after translation:** The slope $$m$$ remains unchanged because vertical shifts do not affect the slope. So, yes, the translated pathway has the same slope as the original. 4. **Translate 4 units downward:** Translating downward by 4 units means subtracting 4 from the y-values. New equation: $$y = mx + b - 4$$ 5. **Compare translations upward and downward:** The only part that changed is the constant term (y-intercept) from $$b$$ to $$b + 5$$ or $$b - 4$$. This is because vertical translations shift the graph up or down without changing the slope. 6. **Reflect across the x-axis:** Reflection across the x-axis changes the sign of the y-values. New equation: $$y = - (mx + b) = -mx - b$$ 7. **Reflect across the y-axis:** Reflection across the y-axis changes the sign of the x-values. New equation: $$y = m(-x) + b = -mx + b$$ 8. **Compare reflections across x-axis and y-axis:** - Reflection across x-axis: $$y = -mx - b$$ - Reflection across y-axis: $$y = -mx + b$$ The difference is in the sign of the y-intercept term. 9. **Transformation changing only y-intercept:** Vertical translations (up or down) change only the y-intercept while keeping the slope the same. 10. **Transformations keeping pathway parallel:** Translations up or down keep the slope $$m$$ unchanged, so the new pathway remains parallel to the original. Reflections change the slope's sign or the equation form, so they do not keep the pathway parallel. **Summary:** - Original: $$y = mx + b$$ - Up 5 units: $$y = mx + b + 5$$ - Down 4 units: $$y = mx + b - 4$$ - Reflect x-axis: $$y = -mx - b$$ - Reflect y-axis: $$y = -mx + b$$