The Exponential Function

When your group graphed the layer counts from Folding Paper to the Moon, the dots hugged the floor and then left the page — a shape no line or parabola makes. Join those dots and you have $f(x)=2^x$: the exponential function. In general,

$f(x)=a^x,a>0,a\ne 1$

is a function — each input gets exactly one output — and thanks to Rational Exponents it now makes sense at every real input, not just whole numbers.

The tell: a constant ratio

A linear function grows by adding the same amount each step; an exponential grows by multiplying by the same factor each step. That is how you unmask one in a table of values: first differences equal means linear, second differences equal means quadratic, but a constant ratio between consecutive outputs means exponential. The values $3,6,12,24$ whisper “ratio 2” — no graph required.

Key properties

For $f(x)=a^x$, every member of the family shares:

• domain: all real numbers; range: $y>0$ — the output is a power of a positive base, and no such power is zero or negative
• $y$-intercept 1, because $a^0=1$
• horizontal asymptote $y=0$ — the graph approaches the axis and never arrives
• increasing everywhere when $a>1$; decreasing everywhere when $0<a<1$ — never flat, which is why $a=1$ is excluded

And the recipe from Transformations of Functions applies without amendment: $y=a\cdot 2^{k(x-d)}+c$ moves this parent exactly as it moved $x^2$ and $\sqrt{x}$. Note where the asymptote goes — it rides the vertical shift up to $y=c$, which drags the range along.

Growth, decay, and the real world

Real situations arrive as initial amount times repeated multiplier: a town growing 3% per year is $P(t)=P_0(1.03{)}^t$; a medication with a half-life of 6 hours is $M(t)=M_0{\left(\frac{1}{2}\right)}^{t/6}$. Questions about the future are answered by substituting into the equation; questions in reverse (“when does it reach 28°C?”) are answered from the graph or by systematic guess-and-check — Using Desmos handles both gracefully.

Check your understanding

A ball dropped from 2 m rebounds to 60% of its previous height on each bounce. How high is bounce four? Each bounce multiplies by 0.6, so $h=2(0.6{)}^4=2(0.1296)\approx 0.26$ m. Sanity check: four multiplications by a number below 1 should leave under an eighth of the original height, and it does.

Whether a situation is genuinely exponential — and what its domain honestly means — is the heart of Double or Nothing, and the same mathematics runs compound interest. Exponential Models Practice covers the full span, from properties to predictions.

Curriculum connection

B1.1

graph, with and without technology, an exponential relation, given its equation in the form $y=a^x$ $(a>0,a\ne 1)$, define this relation as the function $f(x)=a^x$, and explain why it is a function

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B1.4

determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form $f(x)=a^x$ $(a>0,a\ne 1)$, function machines] Sample problem: Graph $f(x)=2^x$, $g(x)=3^x$, and $h(x)=0.5^x$ on the same set of axes. Make comparisons between the graphs, and explain the relationship between the y-intercepts.

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B2.1

distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations) Sample problem: Explain in a variety of ways how you can distinguish the exponential function $f(x)=2^x$ from the quadratic function $f(x)=x^2$ and the linear function $f(x)=2x$.

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B3.3

solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations Sample problem: The temperature of a cooling liquid over time can be modelled by the exponential function $T(x)=60{\left(\frac{1}{2}\right)}^{x/30}+20$, where $T(x)$ is the temperature, in degrees Celsius, and $x$ is the elapsed time, in minutes. Graph the function and determine how long it takes for the temperature to reach 28°C.

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