Square Root Function Transformation Notes
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This document provides information and examples for graphing square root functions, including: - Tips for graphing square root functions such as how the h and k terms translate the function and how the a term forms the shape. - Examples of square root functions with questions asking to identify the minimum/maximum point, domain, range, x-intercepts and y-intercepts. - Explanations of how to find the minimum/maximum point from a graph or equation, noting it is the (h,k) point. - A description of vertex form for square root functions where the a term reflects or dilates, and h and k translate, with the minimum/maximum point again being
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Vertical translations move the graph up or down by adding or subtracting a number from the y-values. Horizontal translations move the graph left or right by adding or subtracting a number from the x-values. Reflections about the x or y-axis flip the graph across that axis by multiplying the x or y-values by -1. Stretching or compressing vertically or horizontally multiplies the y or x-values by a number, scaling the graph proportionally larger or smaller.
Algebra 2. 9.15. Intro to quadratics
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
Functions
This document discusses functions and how to determine if a relation is a function. It defines a function as a relation where each element of the domain maps to exactly one element of the range. It provides examples of tables, coordinate points, and graphs, and explains how to use the vertical line test to determine if a graph represents a function. It also discusses how to determine if a relation is a function algebraically by solving for the output variable and checking if there is only one output for each input.
Graph transformations
The document discusses various types of graph transformations including reflections, translations, and dilations. Reflections involve reflecting the graph across the x or y-axis. Translations involve shifting the graph horizontally or vertically by some value. Dilations involve stretching or compressing the graph by some scale factor along the x or y-axis. For each transformation type, the document provides the formula used to transform the original function f(x) to the transformed function g(x).
CAT Functions and Graphs basics
This document provides an overview of key concepts related to functions and graphs including:
- The definitions of relations and functions and the differences between them
- Common types of functions such as constant, identity, and absolute value functions
- Transformations of graphs including vertical and horizontal shifts, squeezing and stretching
- Properties of even, odd, and inverse functions
- Techniques for finding the domain of a function
- Examples of problems involving functions and their solutions
Piecewise functions
The document discusses piecewise functions, which are functions defined by multiple rules over different parts of the domain. It provides examples of piecewise functions, including how to graph them by applying each rule over the appropriate portion of the domain. It also discusses evaluating piecewise functions at given values, writing piecewise functions based on graphs, and analyzing properties like extrema. Piecewise functions allow modeling of real-world situations using multiple equations each corresponding to a part of the input range.
Functions
Summary of aspects of functions: domain, range, domain restrictions.
Worked out for all elementary mathematical functions: linear, quadratic, etc.
Chapter 1 - What is a Function.pdf
This document defines and provides examples of functions. The key points are:
- A function assigns each element in its domain to exactly one element in its range. The domain is the set of inputs and the range is the set of outputs.
- A function is represented graphically by a curve that passes the vertical line test - no vertical line intersects the curve more than once.
- Functions can be even if f(x) = f(-x), odd if f(x) = -f(-x), or neither. Examples of even, odd, and neither functions are given.
- Transformations of functions including vertical and horizontal shifts, stretching, compressing, and reflections are described along
3.1 Quadratic Functions and Models
Graph a quadratic function from the vertex form
Complete the square to transform a quadratic function to vertex form
Derive the vertex formula
Transformations of functions
This document discusses transformations of functions. It begins by introducing common parent functions like linear, quadratic, absolute value, cubic and square root functions using xy-tables and graphs. It then demonstrates how adding or subtracting values like 1, 5, or 2 to a quadratic function f(x)=x^2 results in a rigid transformation that shifts the graph up, down, left or right but maintains the same basic shape. The document defines rigid transformations as those that only change the position of the graph and lists the three types as vertical shifts, horizontal shifts, and reflections, providing formulas for shifting graphs vertically and horizontally.
2.6 transformations
- The document discusses function transformations including translations, dilations, rotations, and reflections. It focuses on translations, which shift the graph up/down or left/right, and dilations, which shrink or stretch the graph.
- Examples show how to write functions for translations like k(x)=f(x)+3 which shifts the graph up 3 units, and dilations like h(x)=2f(x) which stretches the graph vertically by a factor of 2.
- Questions provide the definition of a function f(x) and ask the reader to write the tables or sketch the graphs for new functions using transformations of f(x).
Deriving the inverse of a function1
The document explains how to derive the inverse of a function. It demonstrates finding the inverse of functions f(x)=2x+5, g(x)=x/7+4, and h(x)=(5x+4)/(x+9) through the following steps: 1) rewrite the original function as y=fx(x), 2) interchange x and y, 3) isolate y to obtain the inverse function f^-1(x). The inverse of f(x) is f^-1(x)=(x-5)/2. The inverse of g(x) is g^-1(x)=7(x-4). The inverse of h(x) is h^-1(x)=(
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Square Root Function Transformation Notes
- 2. NOTES TO REVIEW Please take out the following worksheets/packets to review! Quadratic Transformations
- 3. DO NOW 𝑓 𝑥 = 𝑥 − 2 − 3 Horizontal translation Vertical translation 𝑓 𝑥 = 2 𝑥 + 5 Vertical dilation Horizontal translation 𝑓 𝑥 = 1 2 𝑥 + 2 Vertical dilation Vertical translation 𝑓 𝑥 = − 𝑥 + 1 + 4 Reflection over x-axis Horizontal translation Vertical translation
- 4. TIPS FOR GRAPHING SQUARE ROOT FUNCTIONS The h & k will move the function around the graph. The 𝑎 term is what forms the shape of the function Graph the parent function & then transform all the points OR Use a function table to find the values
- 5. GRAPH SQUARE ROOT FUNCTIONS 𝑓 𝑥 = 𝑥 − 2 − 3 Min/Max Point? Domain? Range? X-Intercept? Y-Intercept?
- 6. GRAPH SQUARE ROOT FUNCTIONS 𝑓 𝑥 = 2 𝑥 + 5 Min/Max Point? Domain? Range? X-Intercept? Y-Intercept?
- 7. GRAPH SQUARE ROOT FUNCTIONS 𝑓 𝑥 = 1 2 𝑥 + 2 Min/Max Point? Domain? Range? X-Intercept? Y-Intercept?
- 8. GRAPH SQUARE ROOT FUNCTIONS 𝑓 𝑥 = − 𝑥 + 1 + 4 Min/Max Point? Domain? Range? X-Intercept? Y-Intercept?
- 9. MINIMUM/MAXIMUM POINT Finding the Min/Max Point from the Graph: •The highest or lowest point of the graph •“Starting” Point Finding the Min/Max Point from the Equation: •The x-coordinate is the horizontal translation (h) •The y-coordinate is the vertical translation (k) •The min/max point is written as (h,k)
- 10. VERTEX FORM 𝑓 𝑥 = 𝑎 𝑥 − ℎ + 𝑘 “a” is the reflection/dilation “h” is the horizontal translation “k” is the vertical translation The min/max point is (h,k) The graph opens ______ if a>0 and opens ______ if a<0. up dow n