Let's go through the exercise step-by-step based on what the questions are asking.

Exercise 14A

1. Graphing Exponential Functions

   • 1a: Draw the graph of $y = (1.7)^x$ for $-4 \leq x \leq 4$.
     • This is an exponential growth function, which increases as $x$ increases because $1.7 > 1$.
   • 1b: Use the graph to solve the equation $(1.7)^x = 4$.
     • Graphically, find the $x$-value where $y = 4$ on the graph of $y = (1.7)^x$.

2. Graphing a Decaying Exponential Function

   • 2a: Draw the graph of $y = (0.6)^x$ for $-4 \leq x \leq 4$.
     • This is an exponential decay function, so it decreases as $x$ increases because $0.6 < 1$.
   • 2b: Use the graph to solve $(0.6)^x = 2$.
     • This will likely involve extrapolating or finding an approximate $x$-value where $y = 2$.

3. Sketching a Constant Function

   • 3: Sketch the graph of $y = 1^x$.
     • Since $1^x = 1$ for all $x$, this is a horizontal line at $y = 1$.

4. True/False Statements

   • 4a: Determine if the graph of $y = a^x$ passes through $(0, 1)$ for all positive real numbers $a$.
     • True, because $a^0 = 1$ for any positive $a$.
   • 4b: Decide if $f(x) = a^x$ is always increasing for $a > 0$.
     • This is true if $a > 1$, false if $0 < a < 1$.
   • 4c: Check if the graph of $y = a^x$, where $a$ is positive, never crosses the $x$-axis.
     • True, as exponential functions do not cross the $x$-axis (they approach but never reach zero).

5. Exploring Transformations of Exponential Functions

   • Given $f(x) = 3^x$, plot three transformations:
     • 5a: $y = f(x)$ (which is $y = 3^x$).
     • 5b: $y = 2f(x)$ (which is $y = 2 \cdot 3^x$), a vertical stretch.
     • 5c: $y = f(x) - 4$ (which is $y = 3^x - 4$), a vertical shift downward.
   • Find the $y$-intercepts and any horizontal asymptotes for each function.

6. Finding Constants for an Exponential Equation

   • 6: Given that the graph of $y = ka^x$ passes through points $(1, 6)$ and $(4, 48)$, solve for $k$ and $a$.
     • Substitute the points into the equation $y = ka^x$ to form a system of equations and solve for $k$ and $a$.

Would you like a specific solution or detailed explanations for any of these questions?

Here are some related questions to consider:

1. How do you determine if an exponential function is increasing or decreasing?
2. What is the effect of vertical and horizontal shifts on exponential functions?
3. How can you solve exponential equations graphically versus algebraically?
4. Why does $y = a^x$ always pass through the point $(0, 1)$ when $a > 0$?
5. How do you interpret the horizontal asymptote in exponential functions?

Tip: Remember that in exponential functions $y = a^x$, the value of $a$ determines the growth or decay behavior.