Transforming Exponential Functions

You already know what $a$, $k$, $d$, and $c$ do to a parabola. They do exactly the same things to an exponential curve, which is the point: transformations are a property of functions, not of parabolas. What changes is which features move, because an exponential has an asymptote and a parabola does not.

The general form

$y=af(k(x-d))+c\text{where }f(x)=b^x$

Parameter	What it does	What to watch
$a$	Vertical stretch by $|a|$; reflects in the $x$-axis when $a<0$	The asymptote does not move
$k$	Horizontal compression by $\frac{1}{|k|}$; reflects in the $y$-axis when $k<0$	The $y$-intercept does not move
$d$	Horizontal translation, right when $d>0$	Inside the bracket, so it works backwards
$c$	Vertical translation, up when $c>0$	The asymptote moves to $y=c$

That last row is the whole difference from quadratics. An exponential function’s graph approaches a horizontal line it never reaches, and only $c$ can move it.

Watching it happen

Start from $y=2^x$, which passes through $(0,1)$ with asymptote $y=0$, and take one step at a time in Using Desmos. Predict each before you press enter:

$y=3(2^x)y=2^{x-4}y=2^x-5y=-2^x$

The third one is the interesting one. Subtracting 5 drags the whole curve down, so the asymptote becomes $y=-5$ — and the graph now crosses the $x$-axis, which $y=2^x$ never does. A transformation changed the number of zeros, and that is not a cosmetic change.

Reading the equation off a graph

Given a graph, work in this order and the algebra stays easy:

1. Find the asymptote. That is $c$, immediately.
2. Subtract it away. What remains behaves like $y=a{b}^{k(x-d)}$ with asymptote zero.
3. Use two points. Substituting gives two equations; dividing one by the other kills $a$ and leaves you solving for the base or for $k$.
4. Check with a third point. If it does not fit, one of your first two readings was off the grid line.

Worked briefly: a curve with asymptote $y=3$ passing through $(0,5)$ and $(1,7)$. Then $c=3$, and $5-3=2$ gives $a=2$ at $x=0$; $7-3=4$ gives $2b=4$, so $b=2$. The function is $y=2(2^x)+3$, and a third point will confirm it.

Why do $a$ and $k$ overlap for exponentials?

Because a horizontal stretch of an exponential is also a vertical stretch: $2^{x-3}=2^{-3}\cdot 2^x=\frac{1}{8}(2^x)$. Sliding the curve sideways is indistinguishable from scaling it vertically — which is true for exponentials and false for parabolas. Two different equations can therefore describe the same curve, and both are correct.

Why this matters beyond the graph

Every growth or decay situation in The Exponential Function arrives with a starting amount and a rate — which is to say, with $a$ and $b$ already decided by the context. The transformations are how you get from the situation to the equation without guessing, and how you recognise, from a graph somebody hands you, what the situation must have been.

Curriculum connection

B2.2

determine, through investigation using technology, the roles of the parameters $a$, $k$, $d$, and $c$ in functions of the form $y=af(k(x-d))+c$, and describe these roles in terms of transformations on the graph of $f(x)=a^x$ $(a>0,a\ne 1)$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) Sample problem: Investigate the graph of $f(x)=3^{x-d}-5$ for various values of $d$, using technology, and describe the effects of changing $d$ in terms of a transformation.

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B2.3

sketch graphs of $y=af(k(x-d))+c$ by applying one or more transformations to the graph of $f(x)=a^x$ $(a>0,a\ne 1)$, and state the domain and range of the transformed functions Sample problem: Transform the graph of $f(x)=3^x$ to sketch $g(x)=3^{-(x+1)}-2$, and state the domain and range of each function.

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B2.5

represent an exponential function with an equation, given its graph or its properties Sample problem: Write two equations to represent the same exponential function with a y-intercept of 5 and an asymptote at $y=3$. Investigate whether other exponential functions have the same properties. Use transformations to explain your observations.

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