2.6.4 Graphing Exponential Functions

Sketching and analyzing graphs of exponential functions, identifying key features, and understanding transformations.

定义

An exponential function is a function of the form \(f(x) = a \cdot b^{x-h} + k\), where \(a\) is a non-zero real number (the vertical stretch/compression factor), \(b > 0\) and \(b \neq 1\) is the base, \(h\) is the horizontal shift, and \(k\) is the vertical shift. The parent exponential function is \(f(x) = b^x\). Key features include: the horizontal asymptote at \(y = k\), the y-intercept at \((0, a \cdot b^{-h} + k)\), domain of all real numbers, and range of \((k, \infty)\) if \(a > 0\) or \((-\infty, k)\) if \(a < 0\). When \(b > 1\), the function is increasing (exponential growth); when \(0 < b < 1\), the function is decreasing (exponential decay). The graph passes through the point \((h, a + k)\) and approaches the asymptote \(y = k\) as \(x \to \pm\infty\).

核心公式

• \(f(x) = a \cdot b^{x-h} + k\)
• \(f(x) = a \cdot e^{r(x-h)} + k\) (natural exponential form)
• \(\text{Horizontal asymptote: } y = k\)
• \(\text{y-intercept: } (0, a \cdot b^{-h} + k)\)
• \(\text{Point on graph: } (h, a + k)\)

易错点

• ⚠️ Confusing the direction of horizontal shifts: \(f(x) = b^{x-h}\) shifts RIGHT by \(h\) units (not left), and \(f(x) = b^{x+h}\) shifts LEFT by \(h\) units
• ⚠️ Forgetting that the horizontal asymptote is \(y = k\) (not \(y = 0\)) when the function is vertically shifted, and incorrectly stating the range as \((0, \infty)\) instead of \((k, \infty)\)
• ⚠️ Misidentifying whether the function represents growth or decay: when \(b > 1\) it's growth (increasing), but when \(0 < b < 1\) it's decay (decreasing); also confusing the effect of negative \(a\) on whether the graph is reflected across the asymptote
• ⚠️ Incorrectly calculating the y-intercept by using \(f(0) = a \cdot b^0 + k = a + k\) without accounting for the horizontal shift, or forgetting to apply the shift parameter \(h\) in the formula