Question: 6. g(x) = 3(x + 2)^2 shift left by 2 units stretch vertically by 3 7. g(x) = -(x - 5)^2 + 1 shift right by 5 units shift up by 1 unit 8. Derek walks to his best friend's house at a rate of 1 block per minute, then turns around and walks home. The graph shows the distance Derek walks in the given amount of time. Write an equation for the graph. Graph: a V-shaped line segment on the top-right, with endpoints at (0, 0) and (20, 0), and vertex at (10, 10); position_hint = top-right. 9. Given the parent function f(x) = x^2, what is the new equation if the function is translated 4 units to the right and 3 units down?

1. **Problem 6:** Given $$g(x) = 3(x + 2)^2$$, apply a shift left by 2 units and stretch vertically by 3. 2. **Step 1:** Understand the transformations. - Shifting left by 2 units means replacing $$x$$ by $$x + 2$$. - Stretching vertically by 3 means multiplying the function by 3. 3. **Step 2:** The original function already has $$x + 2$$ inside the square, so it is shifted left by 2 units. 4. **Step 3:** The function is multiplied by 3, so it is stretched vertically by 3. 5. **Step 4:** The transformed function is $$g(x) = 3(x + 2)^2$$. --- 6. **Problem 7:** Given $$g(x) = -(x - 5)^2 + 1$$, apply a shift right by 5 units and shift up by 1 unit. 7. **Step 1:** The function already has $$(x - 5)$$ inside the square, which means it is shifted right by 5 units. 8. **Step 2:** The $$+1$$ outside the square shifts the graph up by 1 unit. 9. **Step 3:** The function is $$g(x) = -(x - 5)^2 + 1$$. --- 10. **Problem 8:** Derek walks to his friend's house at 1 block per minute, then returns home. The graph is a V-shaped line with endpoints at $$(0,0)$$ and $$(20,0)$$ and vertex at $$(10,10)$$. 11. **Step 1:** The graph is an absolute value function centered at the vertex. 12. **Step 2:** The vertex form of an absolute value function is $$y = a|x - h| + k$$ where $$(h,k)$$ is the vertex. 13. **Step 3:** Here, $$h = 10$$ and $$k = 10$$. 14. **Step 4:** Find $$a$$ using a point, for example $$(0,0)$$: $$0 = a|0 - 10| + 10$$ $$0 = 10a + 10$$ $$10a = -10$$ $$a = -1$$ 15. **Step 5:** The equation is $$y = -1|x - 10| + 10$$ or $$y = -|x - 10| + 10$$. --- 16. **Problem 9:** Given the parent function $$f(x) = x^2$$, translate 4 units right and 3 units down. 17. **Step 1:** Translation right by 4 units means replace $$x$$ by $$x - 4$$. 18. **Step 2:** Translation down by 3 units means subtract 3 from the function. 19. **Step 3:** The new function is $$f(x) = (x - 4)^2 - 3$$. **Final answers:** - Problem 6: $$g(x) = 3(x + 2)^2$$ - Problem 7: $$g(x) = -(x - 5)^2 + 1$$ - Problem 8: $$y = -|x - 10| + 10$$ - Problem 9: $$f(x) = (x - 4)^2 - 3$$