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Let's analyze each part of the question to determine the horizontal and vertical shifts of the given graphs:

### Part (a)
The graph given is:
\[ y = (x - 4)^2 + 10 \]
compared to:
\[ y = x^2 + 10 \]

**Horizontal Shift**: The expression \((x - 4)^2\) indicates a horizontal shift to the right by 4 units.

**Vertical Shift**: Since both equations have the same vertical term \(+10\), there is no vertical shift.

### Part (b)
The graph given is:
\[ y = x^2 + 14 \]
compared to:
\[ y = x^2 + 10 \]

**Horizontal Shift**: There is no change in the \(x\)-term, so no horizontal shift occurs.

**Vertical Shift**: The constant changes from \(10\) to \(14\), indicating a vertical shift upward by 4 units.

### Part (c)
The graph given is:
\[ y = (x - 5)^3 - 5(x - 5) - 4 \]
compared to:
\[ y = x^3 - 5x \]

**Horizontal Shift**: The expressions \((x - 5)^3\) and \(-5(x - 5)\) indicate a horizontal shift to the right by 5 units.

**Vertical Shift**: The constant shifts from \(0\) to \(-4\), indicating a vertical shift downward by 4 units.

### Part (d)
The graph given is:
\[ y = 3(x + 2)^2 - (x + 2) + 14 \]
compared to:
\[ y = 3x^2 - x + 10 \]

**Horizontal Shift**: The expressions involving \((x + 2)\) suggest a horizontal shift to the left by 2 units.

**Vertical Shift**: The constant changes from \(10\) to \(14\), indicating a vertical shift upward by 4 units.

### Summary of shifts:
1. **(a) Horizontal shift**: 4 units to the right, **Vertical shift**: None.
2. **(b) Horizontal shift**: None, **Vertical shift**: 4 units upward.
3. **(c) Horizontal shift**: 5 units to the right, **Vertical shift**: 4 units downward.
4. **(d) Horizontal shift**: 2 units to the left, **Vertical shift**: 4 units upward.

Would you like a more detailed explanation of these shifts or help with another concept? 

Here are 5 related questions to deepen your understanding:
1. What causes a function's graph to shift horizontally?
2. How can you determine the vertical shift of a graph by looking at its equation?
3. How do you recognize when a graph is shifted to the right or left?
4. What is the impact of modifying the constant term in a quadratic equation?
5. How do you combine multiple shifts in both the horizontal and vertical directions?

**Tip**: Horizontal shifts are affected by the values inside the parentheses with the \(x\)-term, while vertical shifts are determined by changes outside the parentheses or in the constant term.

Shift the graph of various functions horizontally and vertically based on the transformations provided.

Explore horizontal and vertical shifts of quadratic and cubic graphs. Understand how changing the values inside and outside of function expressions results in graph transformations. Suitable for students in Grades 10-12.

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