Mathematics 10 — Exercise 4.2 · Q1 (ii)

Full question

Construct a table of values for the following function and use it to draw the graph on a coordinate plane.

Step-by-step solution

1. The vertex occurs where the expression inside the absolute value is zero.

   $$x+3=0$$

2. Equate the corresponding entries.

   $$x+3=0$$

3. Collect the variable terms on the left and constants on the right.

   $$x=0-3$$

4. Simplify the constant calculation.

   $$x=-3$$

5. Divide both sides by the variable coefficient.

   $$x=-3\times \frac{1}{1}$$

6. Reduce the fraction.

   $$x=-3$$

7. Substitute the vertex input; |0| = 0, leaving the vertical shift.

   $$-4=-4$$

8. Calculate this operation before combining the remaining terms.

   $$1\times \left(-6\right)=-6$$

9. Calculate this operation before combining the remaining terms.

   $$-6+3=-3$$

10. Substitute this x into the inside expression.

    $$1\times \left(-6.0\right)+3=-3.0$$

11. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-6.0\right|=5.0$$

12. Calculate this operation before combining the remaining terms.

    $$1\times \left(-5\right)=-5$$

13. Calculate this operation before combining the remaining terms.

    $$-5+3=-2$$

14. Substitute this x into the inside expression.

    $$1\times \left(-5.0\right)+3=-2.0$$

15. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-5.0\right|=2.0$$

16. Calculate this operation before combining the remaining terms.

    $$1\times \left(-4\right)=-4$$

17. Calculate this operation before combining the remaining terms.

    $$-4+3=-1$$

18. Substitute this x into the inside expression.

    $$1\times \left(-4.0\right)+3=-1.0$$

19. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-4.0\right|=-1.0$$

20. Calculate this operation before combining the remaining terms.

    $$1\times \left(-3\right)=-3$$

21. Calculate this operation before combining the remaining terms.

    $$-3+3=0$$

22. Substitute this x into the inside expression.

    $$1\times \left(-3.0\right)+3=0.0$$

23. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-3.0\right|=-4$$

24. Calculate this operation before combining the remaining terms.

    $$1\times \left(-2\right)=-2$$

25. Calculate this operation before combining the remaining terms.

    $$-2+3=1$$

26. Substitute this x into the inside expression.

    $$1\times \left(-2.0\right)+3=1.0$$

27. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-2.0\right|=-1.0$$

28. Calculate this operation before combining the remaining terms.

    $$1\times \left(-1\right)=-1$$

29. Calculate this operation before combining the remaining terms.

    $$-1+3=2$$

30. Substitute this x into the inside expression.

    $$1\times \left(-1.0\right)+3=2.0$$

31. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3-1.0\right|=2.0$$

32. Calculate this operation before combining the remaining terms.

    $$1\times 0=0$$

33. Calculate this operation before combining the remaining terms.

    $$0+3=3$$

34. Substitute this x into the inside expression.

    $$1\times 0.0+3=3.0$$

35. Take its absolute value, multiply by the outside factor, then add the shift.

    $$-4+3\times \left|3+0.0\right|=5.0$$

36. Plot the table points and join them with the two straight arms of the V-shaped graph.

Answer

The final answer is shown below.