Translations of functions

Learning Objectives Identify horizontal and vertical translations from a function's equation. Write the equation of a function that has been translated from a parent function. Graph a translated function by shifting the key points of its parent function. Apply translation rules to various families of functions, including polynomial, trigonometric, exponential, and logarithmic functions. Analyze the effect of translations on a function's key features, such as domain, range, intercepts, and asymptotes. Explain how translations affect the derivative of a function, recognizing that the slope of the tangent line remains unchanged at corresponding points. Ever wonder how video game developers make a character walk across the screen? 🎮 They use translations to shift the...

TermDefinitionExample Parent FunctionThe simplest, un-transformed version of a function in a particular family.f(x) = x² is the parent function for the family of quadratic functions. f(x) = sin(x) is the parent function for sine waves. Family of FunctionsA set of functions whose graphs have the same basic shape, where each function is a transformation of a single parent function.The functions y = x², y = (x-3)², and y = x²+5 are all in the same family. TranslationA transformation that slides every point of a graph the same distance in the same direction without changing its size, shape, or orientation.Shifting the graph of y = x² two units to the right to get y = (x-2)². Vertical TranslationA translation that moves the graph of a function up or down along the y-axis.The graph of g(x) = x³...

Vertical Translation Rule g(x) = f(x) + k To translate the graph of f(x) vertically, add a constant 'k' to the function's output. If k > 0, the graph shifts up by 'k' units. If k < 0, the graph shifts down by |k| units. Horizontal Translation Rule g(x) = f(x - h) To translate the graph of f(x) horizontally, add a constant 'h' to the function's input 'x'. If h > 0, the graph shifts right by 'h' units (e.g., f(x-3) is a shift 3 units right). If h < 0, the graph shifts left by |h| units (e.g., f(x+2) is f(x-(-2)), a shift 2 units left). Combined Translation Rule g(x) = f(x - h) + k This general form combines both horizontal and vertical translations. The graph of f(x) is shifted 'h' units...