Function Face Off SlidesAlgebra II / Pre-Calc FUNCTION FACE-OFF Comparing Exponential and Rational Attributes Exponential \(y = a \cdot b^{x-h} + k\) vs Rational \(y = \frac{a}{x-h} + k\) Meet the Contestants E Exponential Functions Domain: Always all real numbers \((-\infty, \infty)\). Horizontal Asymptote: Always at \(y = k\). Vertical Asymptote: Never exists! R Rational Functions Vertical Asymptote: Found at \(x = h\). Horizontal Asymptote: Found at \(y = k\). Domain: Restricted (\(x \neq h\)). The Common Ground EXPONENTIAL Endless Domain Smooth curves Always defined HORIZONTAL ASYMPTOTES Both function types approach a flat boundary line as \(x\) heads toward \(\infty\). RATIONAL Breaks in Graph Vertical Asymptotes Restricted Domain Visual Analysis \(g(x) = 2^{3x-1}\) Exponential \(h(x) = \frac{2}{3x-1}\) Rational What do they share? In this example, both functions have a Horizontal Asymptote at \(y=0\) because \(k = 0\). h(x) g(x) VA: x=1/3 HA: y=0 SUMMARY Vertical Rational ONLY. Where denom = 0. Horizontal Both types! Controlled by the constant \(k\). Domain Exponential: All Real. Rational: Restricted.
Function Face Off WorksheetFUNCTION FACE-OFF Exponential vs. Rational Attributes Name: Date: Quick Reference Guide Exponential: \(y = a \cdot b^{x-h} + k\) Domain: Always \((-\infty, \infty)\) Horizontal Asymptote: Always at \(y = k\) Never has a Vertical Asymptote! Rational: \(y = \frac{a}{x-h} + k\) Vertical Asymptote: Found at \(x = h\) Horizontal Asymptote: Found at \(y = k\) Domain: Restricted (\(x \neq h\)) Practice Problems The functions \(g(x) = 2^{3x-1}\) and \(h(x) = \frac{2}{3x-1}\) are graphed. Which of the following attributes is the same for both functions? A Horizontal asymptote B Vertical asymptote C Domain D Range For the function \(f(x) = \frac{4}{x-3} + 2\), where is the vertical asymptote located? A \(x = 2\) B \(x = 4\) C \(x = 3\) D \(x = -3\) Which of the following is TRUE about the domain of an exponential function \(y = 2^{x-1} + 5\)? A It is restricted by the horizontal asymptote. B It is all real numbers \((-\infty, \infty)\). C It is limited to values greater than 5. D It is undefined at \(x = 1\). Both \(g(x) = 5^{x-2} + 7\) and \(h(x) = \frac{1}{x-2} + 7\) have which horizontal asymptote? A \(y = 2\) B \(y = -2\) C \(y = 7\) D \(y = 5\) Which function type NEVER has a vertical asymptote? A Rational B Logarithmic C Exponential D Absolute Value As \(x\) approaches negative infinity, the function \(g(x) = 2^x\) approaches \(y = 0\). This value represents the: A Vertical Asymptote B Horizontal Asymptote C y-intercept D Zero of the function What is the domain of the rational function \(h(x) = \frac{5}{x+10}\)? A All real numbers B \(x \neq 10\) C \(x \neq -10\) D \(x > -10\) If \(k = 0\) for both an exponential and rational function, they will both have which x-axis behavior? A They will both cross the x-axis at \(x = 0\). B They will both have a horizontal asymptote at \(y = 0\). C They will both have a vertical asymptote at \(x = 0\). D Neither function will exist on the positive x-axis. For \(f(x) = 2^x + 3\), the range is \(y > 3\). For \(g(x) = \frac{1}{x} + 3\), the range is: A \(y > 3\) B \(y < 3\) C \(y \neq 3\) D All real numbers Comparing \(g(x) = e^x\) and \(h(x) = \frac{2}{x}\), which is a difference between them? A One has a vertical asymptote and the other does not. B They have different horizontal asymptotes. C One is a function and the other is a relation. D They both have restricted domains.
Function Face Off Answer KeyAnswer Key Function Face-Off: Teacher Edition Teacher Reference 10/10 Correct 1 A Both functions have \(k=0\), giving them a Horizontal Asymptote at \(y=0\). Only the rational function has a vertical asymptote. 2 C Set denominator to zero: \(x - 3 = 0 \implies x = 3\). 3 B Exponential functions are defined for all real numbers. They do not have gaps or vertical asymptotes. 4 C The constant \(+7\) shift moves the horizontal asymptote to \(y = 7\). 5 C Exponential functions grow or decay smoothly without vertical breaks. Rational and logarithmic functions have vertical asymptotes. 6 B The flat boundary that a graph approaches as it extends left or right is the horizontal asymptote. 7 C The function is undefined when \(x+10=0\), which is at \(x = -10\). 8 B When \(k=0\), both types hug the x-axis (\(y=0\)) as an asymptote. 9 C Rational range is usually everything except the horizontal asymptote value. 10 A Exponential functions have HA only. Rational functions typically have both HA and VA. Pedagogical Insight Students often confuse the behavior of these two functions because they both "flatten out." Emphasize that for rational functions, flattening happens on both ends of the x-axis, whereas for parent exponential functions, it only happens on one end.
