Graph exponential functions and identify key features including the y-intercept, the horizontal asymptote, domain, range, and whether the function is increasing or decreasing (MA.912.F.1.3, MA.912.AR.5.6).

What this topic is asking

MA.912.F.1 and AR.5 ask you to graph an exponential function $f(x) = a \cdot b^x$ and read its key features: the $y$-intercept, the horizontal asymptote, the domain and range, and whether it shows growth or decay. Because the B.E.S.T. Algebra 1 EOC calculator does not graph, you build the shape from the equation and a few points.

The shape of an exponential

An exponential $f(x) = a \cdot b^x$ has a distinctive curve, not a straight line and not a parabola. It hugs a horizontal asymptote on one end and shoots off steeply on the other. The base decides which way:

• $b > 1$ (growth): low and flat on the left, rising ever more steeply to the right.
• $0 < b < 1$ (decay): high on the left, falling and flattening toward the asymptote on the right.

Key features

• $y$-intercept: $f(0) = a \cdot b^0 = a$. The intercept is always the initial value $a$.
• Horizontal asymptote: $y = 0$ for the basic form. The graph gets arbitrarily close to the $x$-axis but never reaches it.
• Domain: all real numbers (you can raise $b$ to any power).
• Range: $y > 0$ (the output is always positive when $a > 0$).

How the B.E.S.T. EOC examines this topic

• Multiple choice and editing task. Identify the $y$-intercept, the asymptote, or growth versus decay.
• GRID and matching. Match an equation to its curve, or plot the intercept.
• Multiselect. Select all true statements about an exponential graph.

A clarifying idea: an exponential graph's end behavior is its signature, one end flattens against the asymptote, the other end runs away to infinity. That asymmetry is how you tell an exponential from a line (straight) or a parabola (symmetric, turns around).

Why the y-intercept is always a and the asymptote is y = 0

Both features fall out of the form $f(x) = a \cdot b^x$. At $x = 0$, any positive base satisfies $b^0 = 1$, so $f(0) = a \cdot 1 = a$; the starting coefficient is the $y$-intercept, no computation needed. For the asymptote, consider the decay direction: as $x$ grows for $0 < b < 1$ (or as $x$ goes very negative for $b > 1$), $b^x$ becomes a tinier and tinier positive number, halving or shrinking without limit, but it can never actually be zero because a positive base raised to any power stays positive. So $a \cdot b^x$ approaches $0$ from above without ever reaching it, which is exactly what a horizontal asymptote at $y = 0$ means. This also explains the range $y > 0$: the output is a positive coefficient times a positive power, never zero or negative.

Distinguishing growth from decay on sight

On the EOC you often must label a graph or equation as growth or decay quickly. From the equation, look at the base: above $1$ is growth, between $0$ and $1$ is decay. From a graph, look at the direction: rising left to right is growth, falling left to right is decay (with the flat asymptote end on the right for decay, on the left for growth). From a table, check the multiplier between consecutive equally spaced outputs: a factor above $1$ grows, below $1$ decays. All three views agree because they describe the same base, so pick whichever representation the item gives you.

Try this

Q1. What is the $y$-intercept of $f(x) = 7 \cdot 2^x$? [1 point]

• Cue. $f(0) = 7$, so $(0, 7)$.

Q2. Does $f(x) = 6 \cdot (0.4)^x$ grow or decay? [1 point]

• Cue. Base $0.4 < 1$, so it decays.