$y > k$ (if $a > 0$) or $y < k$ (if $a < 0$).

$y = b^{x-2}$ moves the graph right $2$ units (not left). The graph shifts in the same direction as $h$.

When you add a constant, the asymptote moves. Failing to update the asymptote in a sketch loses marks.

Apply horizontal shift inside the exponent first, then dilations, then vertical shift. Order matters when combining.

The graph approaches the asymptote but never touches it.

Question 1

What is y-intercept?

Accepted Answer

Substitute $x = 0$: $y = a b^{-h} + k$.

Question 2

What is horizontal asymptote?

Accepted Answer

Always $y = k$ (the value the function approaches as the exponent goes to negative infinity for $b > 1$, or positive infinity for $b < 1$).

Question 3

What is range?

Accepted Answer

$y > k$ (if $a > 0$) or $y < k$ (if $a < 0$).

Question 4

What is confusing the direction of horizontal translation?

Accepted Answer

$y = b^{x-2}$ moves the graph right $2$ units (not left). The graph shifts in the same direction as $h$.

Question 5

What is forgetting to update the asymptote?

Accepted Answer

When you add a constant, the asymptote moves. Failing to update the asymptote in a sketch loses marks.

Question 6

What is mixing the order of transformations?

Accepted Answer

Apply horizontal shift inside the exponent first, then dilations, then vertical shift. Order matters when combining.

Question 7

What is treating the asymptote as a point on the graph?

Accepted Answer

The graph approaches the asymptote but never touches it.