To solve each part, let's analyze how transformations affect the given point $(6, 4)$ on the graph of $y = p(x)$.

The original function $y = p(x)$ contains the point $(6, 4)$, which means: $p(6) = 4.$

Now, let's examine each part:

(a) $y = p(4x) - 2$

1. Inside the function, $p(4x)$ compresses the function horizontally by a factor of $\frac{1}{4}$.
   • If $x = 6$ for $p(x)$, then for $p(4x)$, we solve $4x = 6 \Rightarrow x = \frac{6}{4} = 1.5$.
2. Outside the function, $-2$ shifts the entire function down by 2.
   • So, $p(6) = 4$ changes to $p(4 \times 1.5) - 2 = 4 - 2 = 2$.

Therefore, the point for (a) is: $(1.5, 2)$

(b) $y = 4p(x) + 13$

1. Outside the function, $4p(x)$ scales the function vertically by a factor of 4.
   • Since $p(6) = 4$, we have $4p(6) = 4 \times 4 = 16$.
2. Outside the function, $+13$ shifts the entire function up by 13.
   • So, $4p(6) + 13 = 16 + 13 = 29$.

Therefore, the point for (b) is: $(6, 29)$

(c) $y = -\left(p(3(x + 3)) - 5\right)$

1. Inside the function, $3(x + 3)$ compresses the function horizontally by $\frac{1}{3}$ and shifts it left by 3.
   • Solving $3(x + 3) = 6$, we get $x + 3 = 2$ or $x = -1$.
2. Inside the function, $-5$ shifts the output of $p(x)$ down by 5.
   • So, $p(6) = 4$ changes to $p(6) - 5 = 4 - 5 = -1$.
3. Outside the function, $-$ reflects the function over the x-axis.
   • Thus, $-(-1) = 1$.

Therefore, the point for (c) is: $(-1, 1)$

(d) $y = 4p\left(-\frac{1}{3}x - 15\right) - 15$

1. Inside the function, $-\frac{1}{3}x - 15$ involves both a horizontal compression by a factor of 3, reflection across the y-axis, and a shift right by 15.
   • Solving $-\frac{1}{3}x - 15 = 6$, we get $-\frac{1}{3}x = 21$, thus $x = -63$.
2. Outside the function, $4p(x)$ scales the function vertically by 4.
   • Since $p(6) = 4$, we have $4 \times 4 = 16$.
3. Outside the function, $-15$ shifts the entire function down by 15.
   • So, $4p(6) - 15 = 16 - 15 = 1$.

Therefore, the point for (d) is: $(-63, 1)$

Summary of Points

• (a) $(1.5, 2)$
• (b) $(6, 29)$
• (c) $(-1, 1)$
• (d) $(-63, 1)$

Would you like more details on any specific part?

Here are some related questions for practice:

1. How does a horizontal compression affect the x-coordinate of a point?
2. What happens to a point when the function is vertically scaled by a factor of 4?
3. How does a reflection across the x-axis affect the coordinates?
4. How would you approach transformations involving multiple steps?
5. Why does adding or subtracting outside the function affect the y-coordinate?

Tip: Always start with inside transformations (affecting x) before moving to outside transformations (affecting y).