1. The problem is to find all integer-coordinate points on the graph of the function $$y = |-x + 3|$$. 2. We start by rewriting the function: $$y = |3 - x|$$. 3. To find points with integer coordinates, we evaluate $$y$$ at integer values of $$x$$ around the given domain (-4 to 4). 4. Calculate: - For $$x = -3$$: $$y = |3 - (-3)| = |6| = 6$$ (point: (-3,6)) - For $$x = -2$$: $$y = |3 - (-2)| = |5| = 5$$ (point: (-2,5)) - For $$x = -1$$: $$y = |3 - (-1)| = |4| = 4$$ (point: (-1,4)) - For $$x = 0$$: $$y = |3 - 0| = 3$$ (point: (0,3)) - For $$x = 1$$: $$y = |3 - 1| = 2$$ (point: (1,2)) - For $$x = 2$$: $$y = |3 - 2| = 1$$ (point: (2,1)) - For $$x = 3$$: $$y = |3 - 3| = 0$$ (point: (3,0)) - For $$x = 4$$: $$y = |3 - 4| = 1$$ (point: (4,1)) 5. The points with integer coordinates on the graph within the given range are: $$(-3,6), (-2,5), (-1,4), (0,3), (1,2), (2,1), (3,0), (4,1)$$. 6. The problem statement's highlighted six points are $$(-3,0), (-2,1), (-1,2), (0,3), (1,2), (2,1)$$ which correspond to the graph of $$y = |-x + 3|$$ only if we check carefully. 7. Recalculate at $$x=-3$$: $$y = |-(-3)+3| = |3+3|=6$$ not 0. So the given points likely correspond to graph $$y = |-x + 3|$$ but points are different. 8. Check the points from the graph description: (-3,0), (-2,1), (-1,2), (0,3), (1,2), (2,1). 9. Let's verify if these satisfy $$y = |-x + 3|$$: - For $$x = -3$$: $$|-(-3)+3| = |3 + 3| = 6$$, but given point is (−3,0) so no. 10. Possibly the function intended is $$y = |-x - 3|$$ or $$y = |-x + a|$$ for some $$a$$. Check $$y = |-x - 3|$$ at $$x = -3$$: $$|3 - 3| = 0$$ matches point (−3,0). 11. Check remaining given points in $$y = |-x - 3|$$: - $$x = -2$$: $$|-(-2) -3| = |2 -3| = 1$$ matches (−2,1) - $$x = -1$$: $$|1 -3| = 2$$ matches (−1,2) - $$x = 0$$: $$|0 - 3| = 3$$ matches (0,3) - $$x = 1$$: $$|-1 - 3| = 4$$ does not match (1,2) 12. The given points best fit $$y = |-x + 3|$$ for points (0,3), (1,2), (2,1), but not all six points. 13. The question asks to click on all six points of integer coordinates on $$y = |-x + 3|$$ among those six points. 14. So the six points given that lie on the graph $$y = |-x + 3|$$ are: $$(-3,0), (-2,1), (-1,2), (0,3), (1,2), (2,1)$$ 15. Verify each by calculation: - $$x = -3$$: $$y = |-(-3)+3| = |3+3| = 6$$ no - $$x = -2$$: $$y=|2+3|=5$$ no - $$x = -1$$: $$y=|1+3|=4$$ no - $$x = 0$$: $$y=|0+3|=3$$ yes - $$x = 1$$: $$y=|-1+3|=2$$ yes - $$x = 2$$: $$y=|-2+3|=1$$ yes 16. Points on the graph are (0,3), (1,2), (2,1) from the six given, so only these three points belong. Final Answer: The points with integer coordinates on $$y = |-x + 3|$$ are (0,3), (1,2), and (2,1).