Skill: Graph exponential functions using transformations

To graph an exponential function in the form

$$f(x)=a b^{x-h}+k,$$

compare it with the parent function $y=b^x$.

• $h$ shifts the graph horizontally:
  • $x-h$ means shift right $h$ units.
  • For example, $x-1$ means right $1$ unit.
• $k$ shifts the graph vertically:
  • $+k$ means shift up $k$ units.
• $a$ changes the vertical shape:
  • $\vert a \vert >1$ gives a vertical stretch.
  • $0< \vert a \vert <1$ gives a vertical compression.
  • If $a<0$, reflect the graph across the $x$-axis.
• The horizontal asymptote is $y=k$.
• The domain of an exponential function is all real numbers.

Example

Graph

$$f(x)=-2\cdot 3^{x-1}+4.$$

Step 1: Identify the parent function

The parent function is

$$y=3^x.$$

Since $3>1$, the parent graph represents exponential growth.

Step 2: Identify the transformations

Rewrite the function in the form $a b^{x-h}+k$:

$$f(x)=\underbrace{-2}_{a}\cdot \underbrace{3^{x-1}}_{b^{x-h}}+\underbrace{4}_{k}.$$

Therefore:

• $x-1$ shifts the graph right 1 unit.
• The factor $-2$ reflects the graph across the $x$-axis and stretches it vertically by a factor of $2$.
• $+4$ shifts the graph up 4 units.

Notice that $x-h$ determines the shift. Thus, $x-1$ means right $1$, not left $1$.

Step 3: Find the horizontal asymptote

The parent function $y=3^x$ has asymptote $y=0$.

Shifting up $4$ units gives the new horizontal asymptote:

$$\boxed{y=4}.$$

Step 4: Transform some key points

Useful points on $y=3^x$ are

$$(-1,\tfrac13),\qquad (0,1),\qquad (1,3).$$

For the new function, use

$$x_{\text{new}}=x_{\text{parent}}+1$$

because the graph shifts right $1$, and

$$y_{\text{new}}=-2y_{\text{parent}}+4.$$

Transform the points:

$$\begin{array}{c|c} \text{Parent point} & \text{New point} \\ \hline (-1,\tfrac13) & (0,\tfrac{10}{3})\\ (0,1) & (1,2)\\ (1,3) & (2,-2) \end{array}$$

Plot these points and draw a smooth exponential curve approaching, but never touching, $y=4$.

Because of the negative coefficient, the graph decreases from left to right.

Step 5: State the important features

For

$$f(x)=-2\cdot 3^{x-1}+4:$$

• Horizontal asymptote: $\boxed{y=4}$

• Domain: $\boxed{(-\infty,\infty)}$

• Range: $\boxed{(-\infty,4)}$

• $y$-intercept:

  Using the point table, when $x=0$,

$$f(0)=\frac{10}{3}.$$

So the $y$-intercept is $\boxed{(0,\frac{10}{3})}$.

• $x$-intercept:

  Set $f(x)=0$:

$$-2\cdot 3^{x-1}+4=0$$ $$3^{x-1}=2$$ $$x=1+\log_3 2.$$

Thus the $x$-intercept is approximately

$$\boxed{(1.63,0)}.$$

The final graph is a decreasing exponential curve with asymptote $y=4$, passing through points such as $(0,\frac{10}{3})$, $(1,2)$, and $(2,-2)$.