Let's break down the effects of $k$ and $h$ on the graphs of the functions $a^x + k$ and $a^{x-h}$, respectively.

1. Effect of $k$ on $a^x + k$:

When you have a function of the form $f(x) = a^x + k$, the parameter $k$ causes a vertical translation of the graph of $a^x$.

• Effect: If $k > 0$, the graph shifts up by $k$ units. If $k < 0$, the graph shifts down by $|k|$ units.
• Key Point: The shape of the graph remains the same (still an exponential curve), but its position on the vertical axis changes.

Example:

• If the base $a$ is greater than 1 (e.g., $a = 2$), the graph of $f(x) = 2^x$ would typically rise steeply as $x$ increases. When you add $k = 3$, the entire graph shifts up by 3 units, so the horizontal asymptote moves from $y = 0$ to $y = 3$.

2. Effect of $h$ on $a^{x-h}$:

In the case of the function $g(x) = a^{x-h}$, the parameter $h$ causes a horizontal translation of the graph of $a^x$.

• Effect: If $h > 0$, the graph shifts to the right by $h$ units. If $h < 0$, the graph shifts to the left by $|h|$ units.
• Key Point: The shape of the graph remains unchanged, but its position along the horizontal axis shifts.

Example:

• If $f(x) = 2^x$, and you modify the function to $g(x) = 2^{x-3}$, this results in a horizontal shift of the graph to the right by 3 units. The horizontal asymptote, previously at $y = 0$, remains at $y = 0$, but the graph shifts rightward.

Similarities between $k$ and $h$:

• Both parameters $k$ and $h$ affect the position of the graph of the exponential function.
• Both $k$ and $h$ do not change the shape of the graph. The function still has exponential growth or decay characteristics.
• They result in translations of the graph (vertical for $k$ and horizontal for $h$).

Differences between $k$ and $h$:

• $k$ causes a vertical shift of the graph, changing the position along the $y$-axis, but does not affect how the function grows or decays with respect to $x$.
• $h$ causes a horizontal shift of the graph, which changes the behavior of the function along the $x$-axis. This can change the function’s domain in terms of its input, but the general shape remains unchanged.

Summary:

• Vertical Translation: $k$ shifts the graph up or down.
• Horizontal Translation: $h$ shifts the graph left or right.

Would you like more details on these shifts or any examples to illustrate this further?

Related questions:

1. How would the graph of $a^x + k$ behave if $a$ were a fraction between 0 and 1 (e.g., $a = 1/2$)?
2. What happens to the graph of $a^{x-h}$ if $h$ is negative?
3. How does a translation of the form $y = a^{x-h} + k$ differ from a separate translation of the graph of $y = a^x$ in both horizontal and vertical directions?
4. How does the horizontal shift affect the domain and range of the function $a^{x-h}$?
5. How would the graph of $y = 2^{x} + 4$ compare to $y = 2^x - 4$?

Tip: Always remember, adding or subtracting $k$ affects the vertical position, while adding or subtracting $h$ inside the exponent affects the horizontal position.