Graphing Exponential Functions

Plot y = a·b^x from a table of values, read its intercept and asymptote, and see how shifts, reflections and stretches move the graph.

Advanced30 min18 problems
Practice Problems

Definition

To graph an exponential function f(x)=a⋅bxf(x) = a \cdot b^x (with b>0b > 0 and b≠1b \neq 1), make a table of values, plot the points and join them with a smooth curve that bends toward a horizontal asymptote.
The two basic shapes (here with a=1a = 1):
xx−2-2−1-100112233
2x2^x0.250.250.50.511224488
(0.5)x(0.5)^x4422110.50.50.250.250.1250.125
  • If b>1b > 1, the graph rises from left to right: almost flat near the x-axis on the left, steeper and steeper on the right.
  • If 0<b<10 < b < 1, the graph falls from left to right: steep on the left, almost flat near the x-axis on the right.
**Key features of y=a⋅bxy = a \cdot b^x:**
  • y-intercept: (0,a)(0, a), because b0=1b^0 = 1
  • Horizontal asymptote: y=0y = 0 (the x-axis); the curve gets closer and closer but never touches it
  • Domain: all real numbers
  • Range: y>0y > 0 when a>0a > 0, and y<0y < 0 when a<0a < 0
Transformations: y=a⋅bx−h+ky = a \cdot b^{x-h} + k
  • hh shifts the graph hh units right (left when hh is negative)
  • kk shifts the graph kk units up (down when kk is negative) and moves the asymptote to y=ky = k
  • a<0a < 0 reflects the graph in the x-axis (before the shift), so it lies below its asymptote
  • a factor aa with ∣a∣>1\lvert a \rvert > 1 stretches the graph vertically, 0<∣a∣<10 < \lvert a \rvert < 1 compresses it
  • Range of the transformed graph: y>ky > k when a>0a > 0, and y<ky < k when a<0a < 0

Try it now

What is the horizontal asymptote of y=2x−7y = 2^x - 7? Enter the number kk in y=ky = k.

Worked Examples

Graph y=3⋅2xy = 3 \cdot 2^x for x=−2x = -2 to x=3x = 3. State the y-intercept, the asymptote, the domain and the range.
1

Compute the values for negative xx

3⋅2−2=3⋅14=0.753 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = 0.75 and 3⋅2−1=3⋅12=1.53 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5 → Points (−2,0.75)(-2, 0.75) and (−1,1.5)(-1, 1.5)
2

Compute the values for x=0x = 0 to 33

3⋅20=33 \cdot 2^0 = 3, 3⋅21=63 \cdot 2^1 = 6, 3⋅22=123 \cdot 2^2 = 12, 3⋅23=243 \cdot 2^3 = 24 → Points (0,3)(0, 3), (1,6)(1, 6), (2,12)(2, 12), (3,24)(3, 24)
3

Plot and join the points

Each step to the right doubles yy; each step to the left halves it → A rising curve, flat on the left and steep on the right
4

Read the key features

a=3a = 3 gives the y-intercept; nothing is added, so k=0k = 0 → y-intercept (0,3)(0, 3), asymptote y=0y = 0, domain all real numbers, range y>0y > 0

Common Mistakes

Keeping the asymptote at y=0y = 0 after a vertical shift, for example for y=2x+3y = 2^x + 3

Why it's wrong: Adding kk moves every point up by kk, including the level the curve approaches. The graph of y=2x+3y = 2^x + 3 never goes below 3.

Correct: For y=a⋅bx−h+ky = a \cdot b^{x-h} + k the asymptote is y=ky = k. For y=2x+3y = 2^x + 3 it is y=3y = 3 and the range is y>3y > 3.

Shifting the wrong way: thinking y=2x−3y = 2^{x-3} moves the graph 3 units left

Why it's wrong: In x−3x - 3 you need a larger xx to get the same exponent as before. The point (0,1)(0, 1) of y=2xy = 2^x is now at (3,1)(3, 1).

Correct: x−hx - h shifts right by hh, x+hx + h shifts left by hh. y=2x−3y = 2^{x-3} moves right 3, y=2x+3y = 2^{x+3} moves left 3.

Reading the y-intercept as aa (or a+ka + k) when the graph is also shifted sideways

Why it's wrong: The y-intercept is (0,a)(0, a) only for y=a⋅bxy = a \cdot b^x. After a horizontal shift the exponent at x=0x = 0 is no longer 0.

Correct: Always substitute x=0x = 0. For y=2x−1+3y = 2^{x-1} + 3: 2−1+3=3.52^{-1} + 3 = 3.5, not 1 and not 4.

Treating −2x-2^x as (−2)x(-2)^x

Why it's wrong: Powers are computed before the minus sign, so −2x=−(2x)-2^x = -(2^x). The base is 2, not −2-2, and (−2)x(-2)^x is not even an exponential function.

Correct: −2x-2^x is the graph of 2x2^x reflected in the x-axis: at x=2x = 2 it gives −4-4, while (−2)2=4(-2)^2 = 4.

Interactive Visual

Function graph

-4-202401020xyy = 0

Tap or hover over the graph to read off values.

y = a·b^(x − h) + kAsymptote

Table of values

x-2-10123
y0.751.5361224

Function graph

-4-3-2-10123402468xy(0, 1)y = 0

Tap or hover over the graph to read off values.

Tap a name to show or hide its graph.

Asymptote

Table of values

x-2-10123
2^x0.250.51248
0.5^x4210.50.250.125

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is the y-intercept of the graph of y=5⋅3xy = 5 \cdot 3^x?

Why It Matters

A graph shows at a glance what a formula hides. Once you can sketch an exponential graph, you can:
  • Recognize exponential data: when a chart of followers, cells or savings curves upward faster and faster, a table of values tells you whether it really multiplies by the same factor each step.
  • Read the starting value and the floor: the y-intercept is where the quantity starts, and the asymptote y=ky = k is the level it can approach but never cross, such as a baseline that is always there.
  • Check a model quickly: a sketch tells you immediately if a formula rises or falls, where it crosses the axes and which values are impossible (outside the range).
  • Build a model from data: two points like (0,3)(0, 3) and (1,6)(1, 6) are enough to find aa and bb and predict the next values.
These skills come back later in this topic in decay, half-life, continuous growth and exponential equations, and after that in logarithms.

Real World Applications

Reading Lab Data on a Graph

Biologists plot cell counts over time. If the points lie on an exponential curve, two readings are enough to write the model and predict later counts.

Example:

A culture counted 100 cells at the start and 200 cells one hour later follows y=100⋅2xy = 100 \cdot 2^x, so after 4 hours the curve is at 100⋅16=1600100 \cdot 16 = 1600 cells.

1Try It Yourself

A cell culture has 200 cells at x=0x = 0 hours and 600 cells at x=1x = 1 hour, and the graph is exponential: y=a⋅bxy = a \cdot b^x.

How many cells does the model predict at x=3x = 3 hours?

Step 1: Write the mathematical expression

Find aa and bb, then evaluate a⋅b3a \cdot b^3:

Video Views with a Baseline

Some quantities have a part that grows exponentially on top of a constant part. The constant part is the vertical shift $k$, and on the graph it is the asymptote.

Example:

A video gets a steady 100 views a day from search, plus a shared part that doubles every day: V(t)=25⋅2t+100V(t) = 25 \cdot 2^t + 100. Going back in time the shared part shrinks toward 0, so the graph approaches V=100V = 100.

2Try It Yourself

Daily views follow V(t)=25⋅2t+100V(t) = 25 \cdot 2^t + 100, where tt is the number of days since the video was shared.

How many views does the video get on day t=4t = 4?

Step 1: Write the mathematical expression

Evaluate 25⋅24+10025 \cdot 2^4 + 100:

Folding Paper

Each fold doubles the thickness of a sheet of paper, so the thickness after $n$ folds is an exponential function whose graph climbs faster and faster.

Example:

A sheet 0.1 mm thick has thickness T(n)=0.1⋅2nT(n) = 0.1 \cdot 2^n mm. After 7 folds that is 0.1⋅128=12.80.1 \cdot 128 = 12.8 mm.

3Try It Yourself

A sheet of paper is 0.1 mm thick, and its thickness after nn folds is T(n)=0.1⋅2nT(n) = 0.1 \cdot 2^n mm.

After how many folds is the stack thicker than 1 mm for the first time?

Step 1: Write the mathematical expression

Make a table of 0.1⋅2n0.1 \cdot 2^n and look for the first value above 1:

Key Takeaways

  • 1To graph y=a⋅bxy = a \cdot b^x, make a table (for example x=−2x = -2 to 33), plot the points and join them with a smooth curve
  • 2If b>1b > 1 the graph rises from left to right; if 0<b<10 < b < 1 it falls; y=bxy = b^x and y=(1/b)xy = (1/b)^x are mirror images in the y-axis
  • 3For y=a⋅bxy = a \cdot b^x: y-intercept (0,a)(0, a), asymptote y=0y = 0, domain all real numbers, range y>0y > 0 when a>0a > 0
  • 4In y=a⋅bx−h+ky = a \cdot b^{x-h} + k: hh shifts right, kk shifts up and moves the asymptote to y=ky = k, and a<0a < 0 puts the graph below the asymptote
  • 5After a shift, find the y-intercept by substituting x=0x = 0; the range is y>ky > k or y<ky < k
  • 6Two points find the equation: aa is the value at x=0x = 0, and dividing the y-values gives a power of bb

Frequently Asked Questions

No. For y=a⋅bx+ky = a \cdot b^x + k, the term a⋅bxa \cdot b^x is never 0, so yy is never equal to kk. The curve gets as close as you like, which is why the range is y>ky > k or y<ky < k and never includes kk.
No. For y=a⋅bx+ky = a \cdot b^x + k, the term a⋅bxa \cdot b^x is never 0, so yy is never equal to kk. The curve gets as close as you like, which is why the range is y>ky > k or y<ky < k and never includes kk.
Four or five are enough: the y-intercept, one or two points on each side, and the asymptote. A table from x=−2x = -2 to x=3x = 3 shows both the flat side and the steep side.
Because (0.5)x=2−x(0.5)^x = 2^{-x}. The value of (0.5)x(0.5)^x at x=3x = 3 equals the value of 2x2^x at x=−3x = -3, so every point (x,y)(x, y) of one graph has the partner (−x,y)(-x, y) on the other.
No. Shifting left or right slides the curve along the asymptote, so it stays at y=ky = k. Only the vertical shift kk moves the asymptote. A horizontal shift does change the y-intercept, though.

Glossary

Table of values
A list of chosen xx values with the matching yy values, used to plot points of a graph
Horizontal asymptote
A horizontal line y=ky = k that the graph approaches more and more closely but never reaches
y-intercept
The point where the graph crosses the y-axis, found by substituting x=0x = 0
Domain
All xx values a function accepts; for every exponential function it is all real numbers
Range
All yy values the function reaches; for y=a⋅bx+ky = a \cdot b^x + k it is y>ky > k (if a>0a > 0) or y<ky < k (if a<0a < 0)
Transformation
A shift, reflection or stretch that moves or reshapes a graph, described by hh, kk and aa in y=a⋅bx−h+ky = a \cdot b^{x-h} + k

Formula Card

Exponential function

y=a⋅bxy = a \cdot b^x

y-intercept $(0, a)$, asymptote $y = 0$, rises if $b > 1$, falls if $0 < b < 1$

Transformed form

y=a⋅bx−h+ky = a \cdot b^{x-h} + k

Shift right $h$, up $k$; asymptote $y = k$; range $y > k$ if $a > 0$, $y < k$ if $a < 0$

Mirror image

(1/b)x=b−x(1/b)^x = b^{-x}

The graphs of $b^x$ and $(1/b)^x$ are reflections of each other in the y-axis

Base from two points

bx2−x1=y2y1b^{x_2 - x_1} = \frac{y_2}{y_1}

Divide the y-values of two points on $y = a \cdot b^x$ to get a power of $b$

Today's challengeOne problem, three tries

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