Question

match each function to its graph.

1. $y = \sqrt3{x - 1}$
2. $y = x^3 - 3$
3. $y = \sqrt3{x + 1} - 3$
4. $y = -\sqrt3{x + 3}$
5. $y = -(x - 1)^3 - 3$
6. $y = \sqrt3{x + 3}$

Answer:

To solve the problem of matching each function to its graph, we analyze the transformations of the parent function \( y = \sqrt[3]{x} \) or \( y = x^3 \):

Function 7: \( y = \sqrt[3]{x - 1} \)

• Transformation: Horizontal shift of \( y = \sqrt[3]{x} \) to the right by 1 unit.
• Key Point: The point \( (0, 0) \) on \( y = \sqrt[3]{x} \) shifts to \( (1, 0) \).
• Graph Match: Look for the cube - root graph shifted right 1 unit.

Function 8: \( y = x^3 - 3 \)

• Transformation: Vertical shift of \( y = x^3 \) down by 3 units.
• Key Point: The point \( (0, 0) \) on \( y = x^3 \) shifts to \( (0, - 3) \).
• Graph Match: Look for the cubic graph shifted down 3 units.

Function 9: \( y=\sqrt[3]{x + 1}-3\)

• Transformation: Horizontal shift of \( y=\sqrt[3]{x}\) left by 1 unit and vertical shift down by 3 units.
• Key Point: The point \( (0,0) \) on \( y = \sqrt[3]{x} \) shifts to \( (-1,-3) \).
• Graph Match: Look for the cube - root graph shifted left 1 and down 3 units.

Function 10: \( y=-\sqrt[3]{x + 3}\)

• Transformation: Horizontal shift of \( y = \sqrt[3]{x} \) left by 3 units and reflection over the \( x \) - axis.
• Key Point: The point \( (0,0) \) on \( y=\sqrt[3]{x}\) shifts to \( (-3,0) \), and the graph is reflected over the \( x \) - axis (so it goes from increasing to decreasing or vice - versa).
• Graph Match: Look for the reflected cube - root graph shifted left 3 units.

Function 11: \( y=-(x - 1)^3-3\)

• Transformation: Horizontal shift of \( y = x^3 \) right by 1 unit, reflection over the \( x \) - axis, and vertical shift down by 3 units.
• Key Point: The point \( (0,0) \) on \( y = x^3 \) shifts to \( (1,-3) \), and the graph is reflected over the \( x \) - axis.
• Graph Match: Look for the reflected cubic graph shifted right 1 and down 3 units.

Function 12: \( y=\sqrt[3]{x + 3}\)

• Transformation: Horizontal shift of \( y=\sqrt[3]{x}\) left by 3 units.
• Key Point: The point \( (0,0) \) on \( y=\sqrt[3]{x}\) shifts to \( (-3,0) \).
• Graph Match: Look for the cube - root graph shifted left 3 units.

Final Matches (assuming standard graph options, the following are typical matches):

• 7. \( y=\sqrt[3]{x - 1}\): Graph with key point \((1,0)\) (cube - root shifted right 1)
• 8. \( y = x^3-3\): Graph with key point \((0, - 3)\) (cubic shifted down 3)
• 9. \( y=\sqrt[3]{x + 1}-3\): Graph with key point \((-1,-3)\) (cube - root shifted left 1 and down 3)
• 10. \( y =-\sqrt[3]{x + 3}\): Graph with key point \((-3,0)\) and reflected (cube - root shifted left 3 and reflected)
• 11. \( y=-(x - 1)^3-3\): Graph with key point \((1,-3)\) and reflected (cubic shifted right 1, reflected, and down 3)
• 12. \( y=\sqrt[3]{x + 3}\): Graph with key point \((-3,0)\) (cube - root shifted left 3)

If we assume the graphs are labeled in a standard way (e.g., let's say the graphs are labeled A - F from top - bottom or left - right as per the image layout):

• 7. \( y=\sqrt[3]{x - 1}\): Let's say Graph (for example, if the first non - labeled graph after the function list is the one with key point \((1,0)\))
• 8. \( y = x^3-3\): Let's say Graph (the one with key point \((0,-3)\))
• 9. \( y=\sqrt[3]{x + 1}-3\): Let's say Graph (the one with key point \((-1,-3)\))
• 10. \( y =-\sqrt[3]{x + 3}\): Let's say Graph (the one with key point \((-3,0)\) and reflected)
• 11. \( y=-(x - 1)^3-3\): Let's say Graph (the one with key point \((1,-3)\) and reflected)
• 12. \( y=\sqrt[3]{x + 3}\): Let's say Graph (the one with key point \((-3,0)\))

Since the exact graph labels are not fully clear from the text - based description of the image, but the above is the process to match each function to its graph. If we assume a typical answer set (this is a common matching answer for such problems):

1. \( y=\sqrt[3]{x - 1}\) matches (for example) to a graph with vertex at \((1,0)\) (say Graph X)

1. \( y = x^3-3\) matches to a graph with vertex at \((0,-3)\) (say Graph Y)

1. \( y=\sqrt[3]{x + 1}-3\) matches to a graph with vertex at \((-1,-3)\) (say Graph Z)

1. \( y =-\sqrt[3]{x + 3}\) matches to a graph with vertex at \((-3,0)\) and reflected (say Graph W)

1. \( y=-(x - 1)^3-3\) matches to a graph with vertex at \((1,-3)\) and reflected (say Graph V)

1. \( y=\sqrt[3]{x + 3}\) matches to a graph with vertex at \((-3,0)\) (say Graph U)

If we take a more concrete approach (assuming the graphs are as per a standard textbook problem):

• 7. \( y=\sqrt[3]{x - 1}\): The graph that has the cube - root shape shifted right 1 unit.
• 8. \( y = x^3-3\): The cubic graph shifted down 3 units.
• 9. \( y=\sqrt[3]{x + 1}-3\): The cube - root graph shifted left 1 and down 3 units.
• 10. \( y =-\sqrt[3]{x + 3}\): The reflected cube - root graph shifted left 3 units.
• 11. \( y=-(x - 1)^3-3\): The reflected cubic graph shifted right 1 and down 3 units.
• 12. \( y=\sqrt[3]{x + 3}\): The cube - root graph shifted left 3 units.

For example, if we consider the answer in a boxed form for a single function (say function 7):

The graph of \( y=\sqrt[3]{x - 1}\) is the cube - root graph shifted right 1 unit. If we assume the correct graph is labeled (for example) as the one with the point \((1,0)\) and the cube - root shape, and if we had to box the label (say the label is, for example, "Graph A" where Graph A has the cube - root graph shifted right 1 unit), then:

\(\boxed{\text{Graph with }(1,0)\text{ and cube - root shape (e.g., Graph A)}}\)

But since the exact graph labels are not clear from the given image description, the above is the general method to solve the function - graph matching problem.