Introduction
An exponential graph can be moved, stretched, compressed, or reflected without losing its basic structure. These changes are useful because they let us build new graphs from a familiar one and track important features, especially the horizontal asymptote.
Begin with the function \(f(x)=ab^x\), where \(b>0\), \(b\ne 1\), and \(a\ne 0\). Its horizontal asymptote is \(y=0\). This tutorial describes how common transformations change the graph of this function.
A General Form for the Transformations
We will represent a combination of vertical scaling, reflections, and horizontal and vertical shifts by the function below. Here \(c\ne 0\), \(\varepsilon\) is either \(1\) or \(-1\), \(h\) is the horizontal shift, and \(k\) is the vertical shift.
The factor \(c\) controls vertical scaling and, when it is negative, reflection across the \(x\)-axis. The sign \(\varepsilon=-1\) reflects the graph horizontally. The number \(h\) shifts the graph horizontally, and \(k\) shifts it vertically.
The formula also describes how individual graph points move. If \((t,f(t))\) is a point on the original graph, its corresponding point on the transformed graph is
To see this, set the input to the original function equal to \(t\): \(\varepsilon(x-h)=t\). Since \(\varepsilon\) is \(1\) or \(-1\), solving gives \(x=h+\varepsilon t\). The output is then \(c f(t)+k\). This point-mapping rule gives a direct way to check the direction and size of every transformation.
Vertical Scaling and Reflection Across the \(x\)-Axis
For \(g(x)=c f(x)\), every output of \(f\) is multiplied by \(c\). If \(|c|>1\), the graph is stretched vertically by a factor of \(|c|\); if \(0<|c|<1\), it is compressed vertically by that factor. If \(c<0\), the outputs also change sign, reflecting the graph across the \(x\)-axis.
For example, \(g(x)=3ab^x\) multiplies every \(y\)-coordinate of \(f(x)=ab^x\) by \(3\). The point \((0,a)\) becomes \((0,3a)\). If the coefficient were \(-3\), that point would instead become \((0,-3a)\), and the graph would be reflected across the \(x\)-axis as well as stretched.
Vertical scaling and reflection across the \(x\)-axis leave the line \(y=0\) fixed. Thus, without a vertical shift, the horizontal asymptote remains \(y=0\).
Horizontal Shifts and Reflections
In \(f(x-h)\), the shift is in the opposite direction from the sign inside the parentheses: \(f(x-h)\) shifts the graph right by \(h\) units when \(h>0\), whereas \(f(x+h)\) shifts it left by \(h\) units. The point-mapping rule explains why: with no reflection, an original input \(t\) moves to \(x=t+h\).
A horizontal reflection replaces \(x\) by \(-x\), giving \(f(-x)=ab^{-x}\). More generally, \(f(-(x-h))\) reflects the graph horizontally and then shifts it right by \(h\) units. In the point-mapping rule, an original input \(t\) moves to \(x=h-t\).
A horizontal reflection reverses which end of the graph approaches the horizontal asymptote. It does not move the asymptote itself. A horizontal shift also leaves the asymptote's height unchanged.
Vertical Shifts and the Horizontal Asymptote
Adding \(k\) to a function moves every point vertically by \(k\) units. The original horizontal asymptote \(y=0\) therefore moves to \(y=k\). This remains true when the function has also been scaled, reflected, or shifted horizontally.
Justification. The graph of \(f(x)=ab^x\) has horizontal asymptote \(y=0\): at the infinite end where \(f(t)\) approaches \(0\), its points approach that line. Under the transformation, an original point \((t,f(t))\) moves to \((h+\varepsilon t,k+c f(t))\). Its vertical coordinate approaches \(k\), because \(f(t)\) approaches \(0\), so \(k+c f(t)\) approaches \(k\). The horizontal coordinate still goes to one of the two infinite ends: \(h+\varepsilon t\) tends to \(+\infty\) or \(-\infty\) as \(t\) does. Thus the transformed graph approaches \(y=k\) at the corresponding end. Finally, \(b^{\varepsilon(x-h)}>0\) for every real \(x\), so the sign of \(g(x)-k=ca b^{\varepsilon(x-h)}\) is the sign of \(ca\). This proves the stated position of the graph relative to its asymptote.
Worked Examples
Worked Example: Stretch, Shift Right, and Shift Up
Start with \(f(x)=2^x\). Stretch vertically by a factor of \(3\), shift right by \(1\), and shift up by \(2\). Write the transformed function and check its asymptote and selected points.
A shift right by \(1\) replaces \(x\) with \(x-1\), and a shift up by \(2\) adds \(2\). The vertical stretch multiplies the function by \(3\). Therefore,
The original graph has points \((0,1)\) and \((1,2)\). Under the shift right, they move to \((1,1)\) and \((2,2)\). Scaling the outputs by \(3\) and shifting up by \(2\) moves these points to \((1,5)\) and \((2,8)\).
Check by substitution: \(g(1)=3\cdot2^0+2=3+2=5\), and \(g(2)=3\cdot2^1+2=6+2=8\). The original asymptote \(y=0\) moves up by \(2\), so the transformed asymptote is \(y=2\). Since \(g(x)-2=3\cdot2^{x-1}>0\), the graph lies above it.
Worked Example: Shift Left and Down
Let \(f(x)=2(1/2)^x\), and shift its graph left by \(2\) units and down by \(3\) units. Find the transformed function and verify its asymptote.
A shift left by \(2\) replaces \(x\) with \(x+2\). A shift down by \(3\) subtracts \(3\). Thus,
The original point \((0,2)\) moves left to \((-2,2)\), then down to \((-2,-1)\). Substitution confirms the result: \(g(-2)=2(1/2)^0-3=2-3=-1\). Two further checks are \(g(-1)=2(1/2)^1-3=1-3=-2\), and \(g(0)=2(1/2)^2-3=1/2-3=-5/2\).
The original horizontal asymptote is \(y=0\). The horizontal shift leaves its height unchanged, and the downward shift moves it to \(y=-3\). Since \(g(x)+3=2(1/2)^{x+2}>0\), the graph lies above the new asymptote. The computed outputs are also all greater than \(-3\), as expected.
Worked Example: Combine Scaling, Reflections, and a Shift
Start with \(f(x)=2^x\). Reflect horizontally, shift right by \(1\), stretch vertically by a factor of \(2\), reflect across the \(x\)-axis, and shift up by \(1\). Find the function and check points and asymptote.
Horizontal reflection and a shift right by \(1\) give the input \(-(x-1)=1-x\). The vertical stretch and \(x\)-axis reflection together multiply the function by \(-2\), and the upward shift adds \(1\). Therefore,
For a point check, the original point \((0,1)\) moves horizontally to \((1,1)\), since \(x=1-0=1\). Its output is stretched and reflected to \(-2\), then shifted up to \(-1\). Substitution gives \(g(1)=-2\cdot2^0+1=-2+1=-1\). The original point \((1,2)\) moves to \((0,-3)\): its new horizontal coordinate is \(1-1=0\), and its new vertical coordinate is \(-2(2)+1=-3\). Substitution confirms \(g(0)=-2\cdot2^1+1=-4+1=-3\).
The vertical shift puts the horizontal asymptote at \(y=1\). Also, \(g(x)-1=-2\cdot2^{1-x}<0\), so the graph lies below it. Both the point checks and the sign relative to the asymptote agree with the transformation rule.
A Common Sign Mistake
A frequent error is to read \(x-h\) as a shift left by \(h\). For instance, the equation \(y=2^{x-3}+4\) does not shift \(2^x\) left by \(3\). The expression \(x-3\) shifts the graph right by \(3\), and the \(+4\) shifts it up by \(4\). The asymptote is \(y=4\), not \(y=0\).
Check the direction using the original point \((0,1)\). A shift right by \(3\) and up by \(4\) sends it to \((3,5)\). Substitution confirms this: \(2^{3-3}+4=1+4=5\). Testing a point often catches a mistaken shift direction before it affects the rest of a graph.
Why These Transformations Matter
Transformations provide a compact way to describe families of exponential graphs. Instead of plotting every value from scratch, we can start with \(f(x)=ab^x\), track a few points using the point-mapping rule, and identify the new asymptote from the vertical shift.
When analyzing a transformed equation, identify the vertical shift first: it gives the horizontal asymptote \(y=k\). Then interpret the input change to determine horizontal movement or reflection, and the multiplier to determine vertical scaling or reflection. Keep the sign of the coefficient in view: it tells you whether the graph is above or below its asymptote.
Check Your Understanding
For each question, identify the transformation or feature requested. Where asked, check your conclusion by substitution.
- For \(g(x)=4\cdot 3^{x-2}-5\), state the horizontal and vertical shifts and the horizontal asymptote.
- Does \(2^{x+3}\) shift the graph of \(2^x\) left or right, and by how many units?
- For \(g(x)=-\frac12\,2^x+3\), state the vertical scale, reflection, asymptote, and whether the graph lies above or below it.
- Write a function obtained by reflecting \(f(x)=5^x\) across the \(y\)-axis, shifting it left by \(1\), and shifting it down by \(2\).
- For \(g(x)=2\cdot 4^{-(x-1)}+6\), use the general transformation rule to state the asymptote and whether the graph lies above or below it.
