Graph exponential functions that model growth and decay and identify key features, including the y-intercept and the asymptote, and determine the domain and range (TEKS A.3C, A.9A, A.9F).

What this topic is asking

TEKS A.3C, A.9A, and A.9F ask you to graph an exponential function $f(x) = ab^x$ and identify its key features, the $y$-intercept and the asymptote, and to determine its domain and range. The defining feature of an exponential graph is the horizontal asymptote, which sets it apart from lines and parabolas and is the most-tested idea in this category.

The y-intercept is the initial value

Every exponential $f(x) = ab^x$ crosses the $y$-axis at $(0, a)$, because $b^0 = 1$ makes $f(0) = a$. So the $y$-intercept is the initial value, the same $a$ from the growth or decay model. For $f(x) = 5(2)^x$, the $y$-intercept is $(0, 5)$.

The horizontal asymptote

An exponential function has a horizontal asymptote, a line the curve gets ever closer to but never reaches. For the basic $f(x) = ab^x$, the asymptote is $y = 0$ (the $x$-axis).

• For growth ($b > 1$), the curve falls toward $y = 0$ as $x$ goes to negative infinity and rises steeply as $x$ increases.
• For decay ($0 < b < 1$), the curve falls toward $y = 0$ as $x$ increases.

The asymptote is why the range is $y > 0$ (when $a > 0$): the output is always positive but never actually 0.

Growth and decay shapes

The base sets the direction the curve bends.

Domain and range

For an exponential function, the domain is all real numbers (any exponent is allowed), and the range is $y > 0$ when $a > 0$, because the curve never reaches the asymptote $y = 0$. This contrasts with a quadratic, whose range is bounded by the vertex, and a line, whose range is all reals.

How STAAR examines this topic

• Multiple choice. Find the $y$-intercept (the initial value $a$), identify the asymptote, or match a graph to growth or decay.
• Inline choice. Choose growth or decay, the asymptote, and whether the curve approaches from above or below.
• Hot spot. Select points on an exponential graph or identify the asymptote line.

A clarifying idea is that the asymptote is the floor (or ceiling) the curve never crosses: because $b^x$ is always positive, multiplying by $a > 0$ keeps every output positive, so the graph hugs $y = 0$ without touching it, which is the geometric meaning of the range $y > 0$.

How an exponential graph differs from a parabola

Students sometimes confuse an exponential curve with one branch of a parabola, but they behave differently and the test exploits this. A parabola is symmetric about its axis and turns around at a vertex, so it eventually comes back; an exponential curve is not symmetric and never turns around, growing or decaying without bound in one direction while flattening toward its asymptote in the other. A parabola can dip below the $x$-axis (its range is bounded by the vertex), but a basic exponential $ab^x$ with $a > 0$ stays entirely above the asymptote. Recognizing the one-directional, asymptote-hugging shape is how you match an exponential to its graph rather than to a quadratic.

Comparing growth rates on a graph

When two exponential functions are graphed together, the one with the larger base rises faster and overtakes the other, and the one with the larger coefficient $a$ starts higher (greater $y$-intercept). On a comparison item, read the $y$-intercepts to order the starting values and compare the bases to see which curve will dominate as $x$ increases. This is the graphical version of the linear-versus-exponential idea: an exponential with $b > 1$ eventually outpaces any line, because steepness keeps increasing rather than staying constant.

Try this

Q1. State the $y$-intercept and asymptote of $f(x) = 6(0.5)^x$. [1 point]

• Cue. $y$-intercept $(0, 6)$; asymptote $y = 0$.

Q2. Is $f(x) = 3(1.4)^x$ growth or decay, and what is its range? [1 point]

• Cue. Growth ($b > 1$); range $y > 0$.