Question

1. $y = \sqrt{x - 2} + 5$
2. $y = \sqrt{x + 2} - 3$
3. $y = \sqrt3{x + 1} - 4$
4. $y = \sqrt3{x - 1} - 1$

sketch the graph of each function.

1. $y = \sqrt{x} + 5$
2. $y = \sqrt{x} - 2$
3. $y = 3 + \sqrt{x}$
4. $y = \sqrt{x} + 4$

Answer:

To sketch the graph of each function, we can use the transformation of the parent function \( y = \sqrt{x} \) or \( y = \sqrt[3]{x} \). Here, we'll focus on the square root functions (5 - 8) and cube root functions (1 - 4).

Function 5: \( y = \sqrt{x} + 5 \)

The parent function is \( y = \sqrt{x} \) (domain: \( x \geq 0 \), range: \( y \geq 0 \)).

• Transformation: Vertical shift up by 5 units.
• Key Points:
• For \( x = 0 \): \( y = \sqrt{0} + 5 = 5 \) (point: \( (0, 5) \)).
• For \( x = 1 \): \( y = \sqrt{1} + 5 = 6 \) (point: \( (1, 6) \)).
• For \( x = 4 \): \( y = \sqrt{4} + 5 = 7 \) (point: \( (4, 7) \)).
• Graph: Plot these points and draw a curve starting at \( (0, 5) \) and increasing slowly, matching the shape of \( y = \sqrt{x} \) but shifted up.

Function 6: \( y = \sqrt{x} - 2 \)

Parent function: \( y = \sqrt{x} \).

• Transformation: Vertical shift down by 2 units.
• Key Points:
• For \( x = 0 \): \( y = \sqrt{0} - 2 = -2 \) (point: \( (0, -2) \)).
• For \( x = 1 \): \( y = \sqrt{1} - 2 = -1 \) (point: \( (1, -1) \)).
• For \( x = 4 \): \( y = \sqrt{4} - 2 = 0 \) (point: \( (4, 0) \)).
• Graph: Plot these points and draw a curve starting at \( (0, -2) \) and increasing, similar to \( y = \sqrt{x} \) but shifted down.

Function 7: \( y = 3 + \sqrt{x} \) (or \( y = \sqrt{x} + 3 \))

Parent function: \( y = \sqrt{x} \).

• Transformation: Vertical shift up by 3 units.
• Key Points:
• For \( x = 0 \): \( y = \sqrt{0} + 3 = 3 \) (point: \( (0, 3) \)).
• For \( x = 1 \): \( y = \sqrt{1} + 3 = 4 \) (point: \( (1, 4) \)).
• For \( x = 4 \): \( y = \sqrt{4} + 3 = 5 \) (point: \( (4, 5) \)).
• Graph: Plot these points and draw a curve starting at \( (0, 3) \) and increasing, shifted up from \( y = \sqrt{x} \).

Function 8: \( y = \sqrt{x} + 4 \)

Parent function: \( y = \sqrt{x} \).

• Transformation: Vertical shift up by 4 units.
• Key Points:
• For \( x = 0 \): \( y = \sqrt{0} + 4 = 4 \) (point: \( (0, 4) \)).
• For \( x = 1 \): \( y = \sqrt{1} + 4 = 5 \) (point: \( (1, 5) \)).
• For \( x = 4 \): \( y = \sqrt{4} + 4 = 6 \) (point: \( (4, 6) \)).
• Graph: Plot these points and draw a curve starting at \( (0, 4) \) and increasing, shifted up from \( y = \sqrt{x} \).

Cube Root Functions (1 - 4)

For cube root functions (e.g., \( y = \sqrt[3]{x + 1} - 4 \), \( y = \sqrt[3]{x - 1} - 1 \)):

• The parent function is \( y = \sqrt[3]{x} \) (domain: all real \( x \), range: all real \( y \)).
• Transformations:
• Horizontal shift: \( \sqrt[3]{x - h} \) shifts right by \( h \); \( \sqrt[3]{x + h} \) shifts left by \( h \).
• Vertical shift: \( \sqrt[3]{x} + k \) shifts up by \( k \); \( \sqrt[3]{x} - k \) shifts down by \( k \).

For example, \( y = \sqrt[3]{x + 1} - 4 \):

• Shift \( y = \sqrt[3]{x} \) left by 1 unit, then down by 4 units.
• Key points: \( (-1, -4) \) (vertex), \( (0, -3) \) (since \( \sqrt[3]{0 + 1} - 4 = -3 \)), \( (7, -2) \) (since \( \sqrt[3]{7 + 1} - 4 = 2 - 4 = -2 \)).

General Sketching Tips

1. Identify the parent function (square root or cube root).
2. Determine horizontal/vertical shifts, stretches, or reflections.
3. Plot key points (vertex, intercepts, points from the parent function shifted).
4. Draw the curve to match the shape of the parent function, adjusted for transformations.

For the grid graphs (5 - 8), use the provided coordinate planes to plot the key points and draw the curve. For example, for \( y = \sqrt{x} + 5 \), mark \( (0, 5) \), \( (1, 6) \), \( (4, 7) \), and connect them smoothly.