9.2 Graphs of Exponential Functions

9.2 Graphs of Exponential Functions

Table 11 includes graphing exponential functions as its own skill, not only writing y = a b^t from a story. On AAF you may be shown two curves and asked which equation matches, or given y = 3^{x − 2} + 1 and asked for the intercept, the asymptote, or a point that must lie on the graph. This section is that picture: parent y = a^x, growth versus decay, the intercept (0, 1), the horizontal asymptote y = 0, domain and range, and the shift y = a^{x − h} + k.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

Parent graphs y = a^x

For a constant base a > 0 with a ≠ 1, the parent exponential is:

y = a^x

Every such graph shares three features you should recite before looking at options:

1. Y-intercept (0, 1), because a^0 = 1.
2. Horizontal asymptote y = 0 (the x-axis). As the graph decays toward 0 it never crosses the x-axis.
3. Domain all real numbers and range y > 0.

The base decides the direction:

y = 2^x and y = (1/2)^x are reflections of each other across the y-axis, because (1/2)^x = 2^{−x}. The decay graph is the growth graph with x replaced by −x.

Table of values you should know by sight

Skills Insight 200–236 may simply ask you to evaluate 2^5 = 32 or 3^4 = 81 without a picture. On a graph item, those same points are the dots the CAT is describing. Notice that every column hits y = 1 at x = 0. If a pictured graph misses (0, 1) it is not an unshifted y = a^x.

Domain, range, and the asymptote

For y = a^x with a > 0, a ≠ 1:

• Domain: all real x. Negative exponents are reciprocals, not a ban on negative inputs: 2^{−3} = 1/8.
• Range: y > 0. Outputs are never zero and never negative. There is no x-intercept.
• Horizontal asymptote: y = 0. For growth a > 1, the graph approaches 0 as x → −∞. For decay 0 < a < 1, it approaches 0 as x → +∞.

Do not steal domain language from rationals or radicals. An exponential a^x is defined at every real x; it does not require x > 0. The outputs stay positive. Swapping domain and range is a common wrong choice: “domain x > 0, range all reals” is the story of a logarithm, taught next in Logarithms and Logarithmic Properties, because exponential and log are inverse families.

General transformations of functions — vertical stretch, reflect over an axis — are in Transformations and Interpreting Functions in Context. Stay on the exponential-specific features here: intercept (0, 1) until you shift, and a horizontal asymptote, not a vertical one.

Transformations y = a^{x − h} + k

The shifted graph

y = a^{x − h} + k

starts from y = a^x and:

• replaces x by x − h, which shifts right h (left if h is negative);
• then adds k, which shifts up k (down if k is negative).

Consequences AAF actually tests:

• The old intercept (0, 1) moves. The new graph passes through (h, 1 + k), because when x = h you get a^0 + k = 1 + k.
• The horizontal asymptote moves with the vertical shift: y = k, not y = 0.
• Domain is still all reals. Range is y > k if a^x is added to k (the parent range y > 0 lifted by k).

A vertical stretch y = c · a^x with c > 0 changes the intercept to (0, c) and leaves the asymptote at y = 0. AAF may mix a stretch with a shift: y = 2 · 3^{x} + 4 has intercept (0, 6) and asymptote y = 4.

Worked example: y = 3^{x − 2} + 1

Start with y = 3^x. Shift right 2 and up 1.

• Asymptote: y = 1.
• Special point: at x = 2, y = 3^0 + 1 = 2, so the graph goes through (2, 2).
• Y-intercept: plug x = 0 to get y = 3^{−2} + 1 = 1/9 + 1 = 10/9. The intercept is (0, 10/9), not (0, 1).
• Range: y > 1.

Wrong choices treat the −2 as a vertical slide, report asymptote y = 0 still, or move left 2 because of the minus sign. Inside the exponent, x − 2 is a right shift, the same convention as f(x − 2) in the functions chapter.

Another shift: y = 2^{x + 3} − 4 is left 3 and down 4. Asymptote y = −4. Point (−3, 1 − 4) = (−3, −3). Range y > −4. The graph can now sit below the x-axis because the whole parent was dropped 4 units; the parent 2^x could not, but the shifted graph can.

Comparing two exponential graphs

Higher bands want fluency that includes which curve is which. For x > 0, a larger base greater than 1 grows faster:

• At x = 2, 2^2 = 4 and 4^2 = 16, so y = 4^x is above y = 2^x.
• At x = 0 both equal 1, so they share the intercept.
• At x < 0 the comparison flips for bases greater than 1: 4^{−1} = 1/4 while 2^{−1} = 1/2, so y = 4^x is below y = 2^x to the left of 0 (closer to the asymptote).

Also 4^x = (2^2)^x = (2^x)^2, so for x > 0 the 4-base graph is the square of the 2-base graph’s height. That identity is a comparison tool, not a new family.

Decay versus growth on the same axes: y = 5^x grows; y = 5^{−x} decays, and 5^{−x} = (1/5)^x. If one pictured graph is the left-right flip of the other, the equations are a^x and a^{−x} (or a^x and (1/a)^x). Do not invent a different asymptote; both still have y = 0 until a + k appears.

Reading a graph without a printed picture

CAT items may describe the graph in words: “passes through (0, 1) and (1, 5), and approaches the x-axis as x decreases.” Translate:

1. (0, 1) matches parent y = a^x.
2. (1, 5) means a^1 = 5, so a = 5 and y = 5^x.
3. Approaching the x-axis as x decreases is growth (a > 1).

If the graph passes through (0, 3) instead, it is not parent a^x. Candidates include y = 3 · a^x (stretch) or y = a^x + 2 (shift, but then the asymptote is y = 2, not the x-axis). Use the asymptote to tell stretch from shift: stretch keeps y = 0; a vertical shift moves the asymptote.

A graph through (2, 1) with asymptote y = 0 is consistent with y = a^{x − 2}: the parent intercept has slid to x = 2. Then one extra point pins down a.

Handheld calculators are not allowed except with an approved accommodation (College Board calculator policy). Sketch from the table of values rather than plotting pixel-by-pixel. Once you can name intercept, asymptote, domain, range, and the shift y = a^{x − h} + k, you have the graphing skill. Logs as inverse graphs — reflection across y = x — are in the next section.