University:
Amherst College
Course:
MATH 105 | Calculus with Algebra
Academic year:

2020
Views:

41

Pages:

2
Author:

Hallie Randolph

Functions: Parent Functions, Characteristics of Functions and Transformations Comparisons of graphs to their parent functions… Parent Functions There are several FAMILIES (types) of functions that are commonly studied in mathematics. The most basic of these function types is called the “PARENT FUNCTION” of the family. These are Parent Functions that you should know. Linear (y = x) Absolute Value (y=|x|) Quadratic (y = x2) Cubic (y = x3) Rational (y = 1/x) Square Root (y = √ ) Cube Root (y = √ ) Exponential (y = bx) Logarithmic (y = logbx) Special Characteristics of Functions 1. Domain – The set of all inputs (x-values) that “work” in the function 2. Range - The set of all outputs (y-values) that are possible for the function 3. Extrema – Maximum and minimum points on a graph 4. Zero (X-Intercept) – The points at which a graph crosses the x-axis 5. Y-Intercept – The point at which a graph crosses the y-axis 6. Vertex – (Parabolas and Absolute Values) – The point at which the graph changes increasing/decreasing 7. Interval of Increase – The x-values for which the graph goes up (from left to right) 8. Interval of Decrease – The x-values for which the graph goes down (from left to right) 9. Asymptote – A guideline that a graph approaches and eventually follows alongside. Transformations There are several different standard transformations that you have studied over your math education. These include vertical and horzontal shifts (translations), vertical and horizontal stretches/shrinks, and reflections. The following are examples of single standard reflections and what happens to the graph (and coordinates). Given the parent function y = f(x), 1. Vertical Translation (shift) Y = f(x) + 2 2. Horizontal Translation (shift) Y = f(x+2) - The graph moves up 2 - The y-value for each point is increased 2 - The graph moves left 2 (does the opposite direction) - The x-value for each point is decreased by 2 3. Vertical Stretch/Shrink 4. Horizontal Shrink/Stretch Y = 2f(x) - The graph is stretched vertically by a factor of 2 - The y-value of each point is multiplied by 2 5. Vertical Reflection Y = -f(x) - The graph is reflected (flipped) vertically - The graph is reflected across the x-axis - The y-values change sign (+/-) Y = f(2x) - The graph is shrunk (squished from the side) by a factor of 2 - The x-values for the graph are cut in half (divided by 2) 6. Horizontal Reflection Y = f(-x) - The graph is reflected (flipped) horizontally - The graph is reflected across the y-axis - The x-values change sign (+/-)

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