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Section 2.2 A Catalogue of Essential Functions V63.0121.021/041, Calculus I New York University September 8, 2010Announcements First WebAssign-ments are due September 13 First written assignment is due September 15 Do the Get-to-Know-You survey for extra credit! . . . . . .
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Announcements First WebAssign-ments are due September 13 First written assignment is due September 15 Do the Get-to-Know-You survey for extra credit! . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 2 / 31
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Objectives: A Catalog of Essential Functions Identify different classes of algebraic functions, including polynomial (linear, quadratic, cubic, etc.), polynomial (especially linear, quadratic, and cubic), rational, power, trigonometric, and exponential functions. Understand the effect of algebraic transformations on the graph of a function. Understand and compute the composition of two functions. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 3 / 31
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What is a function?DefinitionA function f is a relation which assigns to to every element x in a set Da single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { y | y = f(x) for some x } is called the range of f. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 4 / 31
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Classes of Functions linear functions, defined by slope an intercept, point and point, or point and slope. quadratic functions, cubic functions, power functions, polynomials rational functions trigonometric functions exponential/logarithmic functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 5 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 6 / 31
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Linear functionsLinear functions have a constant rate of growth and are of the form f(x) = mx + b. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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Linear functionsLinear functions have a constant rate of growth and are of the form f(x) = mx + b.ExampleIn New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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Linear functionsLinear functions have a constant rate of growth and are of the form f(x) = mx + b.ExampleIn New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled.AnswerIf x is in miles and f(x) in dollars, f(x) = 2.5 + 2x . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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ExampleBiologists have noticed that the chirping rate of crickets of a certainspecies is related to temperature, and the relationship appears to bevery nearly linear. A cricket produces 113 chirps per minute at 70 ◦ Fand 173 chirps per minute at 80 ◦ F.(a) Write a linear equation that models the temperature T as a function of the number of chirps per minute N.(b) What is the slope of the graph? What does it represent?(c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 8 / 31
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Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 So an equation for T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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Solution The point-slope form of the equation for a line is appropriate here: If a line passes through (x0 , y0 ) with slope m, then the line has equation y − y0 = m(x − x0 ) 80 − 70 10 1 The slope of our line is = = 173 − 113 60 6 So an equation for T and N is 1 1 113 T − 70 = (N − 113) =⇒ T = N − + 70 6 6 6 37 If N = 150, then T = + 70 = 76 1 ◦ F 6 6 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 10 / 31
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Quadratic functions take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 11 / 31
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Quadratic functions take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. Cubic functions take the form f(x) = ax3 + bx2 + cx + d . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 11 / 31
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ExampleA parabola passes through (0, 3), (3, 0), and (2, −1). What is theequation of the parabola? . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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ExampleA parabola passes through (0, 3), (3, 0), and (2, −1). What is theequation of the parabola?SolutionThe general equation is y = ax2 + bx + c. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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ExampleA parabola passes through (0, 3), (3, 0), and (2, −1). What is theequation of the parabola?SolutionThe general equation is y = ax2 + bx + c. Each point gives anequation relating a, b, and c: 3 = a · 02 + b · 0 + c −1 = a · 22 + b · 2 + c 0 = a · 32 + b · 3 + c . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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ExampleA parabola passes through (0, 3), (3, 0), and (2, −1). What is theequation of the parabola?SolutionThe general equation is y = ax2 + bx + c. Each point gives anequation relating a, b, and c: 3 = a · 02 + b · 0 + c −1 = a · 22 + b · 2 + c 0 = a · 32 + b · 3 + cRight away we see c = 3. The other two equations become −4 = 4a + 2b −3 = 9a + 3b . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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Solution (Continued)Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6b . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6bSubtract these two and we have −6 = −6a =⇒ a = 1. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6bSubtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have −12 = 12 + 6b =⇒ b = −4 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)Multiplying the first equation by 3 and the second by 2 gives −12 = 12a + 6b −6 = 18a + 6bSubtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have −12 = 12 + 6b =⇒ b = −4So our equation is y = x2 − 4x + 3 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 14 / 31
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Whole number powers: f(x) = xn . 1 negative powers are reciprocals: x−3 = 3 . x √ fractional powers are roots: x1/3 = 3 x. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 15 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 16 / 31
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DefinitionA rational function is a quotient of polynomials.Example x3 (x + 3)The function f(x) = is rational. (x + 2)(x − 1) . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 17 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 18 / 31
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Sine and cosine Tangent and cotangent Secant and cosecant . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 19 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 20 / 31
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exponential functions (for example f(x) = 2x ) logarithmic functions are their inverses (for example f(x) = log2 (x)) . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 21 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 22 / 31
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Transformations of FunctionsTake the squaring function and graph these transformations: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1 . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 23 / 31
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Transformations of FunctionsTake the squaring function and graph these transformations: y = (x + 1)2 y = (x − 1)2 y = x2 + 1 y = x2 − 1Observe that if the fiddling occurs within the function, a transformationis applied on the x-axis. After the function, to the y-axis. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 23 / 31
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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal ShiftsSuppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units to the left . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Now try these y = sin (2x) y = 2 sin (x) y = e−x y = −ex . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 25 / 31
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Scaling and flippingTo obtain the graph of y = f(c · x), scale the graph of f by c y = c · f(x), scale the graph of f by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flippingTo obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flippingTo obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a If c < 0, the scaling includes a . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flippingTo obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a compression If c < 0, the scaling includes a . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flippingTo obtain the graph of y = f(c · x), scale the graph of f horizontally by c y = c · f(x), scale the graph of f vertically by c If |c| < 1, the scaling is a compression If c < 0, the scaling includes a flip . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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OutlineLinear functionsOther Polynomial functionsOther power functionsRational functionsTrigonometric FunctionsExponential and Logarithmic functionsTransformations of FunctionsCompositions of Functions . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 27 / 31
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Composition is a compounding of functions insuccession g . ◦f . x . f . . g . . g ◦ f)(x) ( f .(x) . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 28 / 31
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ComposingExampleLet f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 29 / 31
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ComposingExampleLet f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 29 / 31
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DecomposingExample √Express x2 − 4 as a composition of two functions. What is itsdomain?Solution √We can write the expression as f ◦ g, where f(u) = u andg(x) = x2 − 4. The range of g needs to be within the domain of f. Toinsure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2. . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 30 / 31
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Summary There are many classes of algebraic functions Algebraic rules can be used to sketch graphs . . . . . .V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 31 / 31
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