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Here are a few recommended readings before getting started with this lesson.
The graph of the parent function y=sqrt(x) and the graph of the radical function y=asqrt(x-h)+k are drawn on the same coordinate plane.
The graph of the function y=sqrt(x-h)+k is shown in the coordinate plane. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
A translation of a function is a transformation that shifts a graph vertically or horizontally. As with the graph of any other function, a vertical translation of the graph of a radical function is achieved by adding some number to every output value of the function rule. Consider the parent function y=sqrt(x). cc Function & Vertical Translation & bykUnits y=sqrt(x) & y=sqrt(x)+k If k is a positive number, the translation is performed upwards. Conversely, if k is negative, the translation is performed downwards. If k=0, then there is no translation. This transformation can be shown on a coordinate plane.
Graph:
Graph:
Graph:
The graphs of the rational function y=sqrt(x) and a vertical or horizontal translation are shown in the coordinate plane.
The graph of the radical function y=asqrt(cx) is shown in the coordinate plane. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
Suppose that a function is horizontally or vertically stretched/shrunk, and that the graphs of the transformed and the original function are both drawn on the same coordinate plane. Then, the values of a or c can be found by following these procedures.
Finding a | Select two points with the same x-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of a is the quotient of the y-coordinate of Q and the y-coordinate of P. |
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Finding c | Select two points with the same y-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of c is the quotient of the x-coordinate of P and the x-coordinate of Q. |
After mastering vertical and horizontal translations of radical functions, Ignacio is having a hard time understanding vertical and horizontal stretches and shrinks of this type of function. He asked for some help from his very good friend Jordan.
In order to help him, Jordan asks Ignacio to consider the following function. f(x)=sqrt(x-1)+3 She then tells him to complete the next two exercises.
Graph:
Graph:
The graph of the parent function y=sqrt(x) is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.
Ignacio considers the following radical function. f(x)=2sqrt(x-2)-1 He wants to answer two practice exercises about reflection of radical functions.
Graph:
Graph:
Ignacio confidently states that he can now solve any exercise about transformations of radical functions. Jordan skeptically challenges her friend to solve an exercise that combines transformations.
Equation: y=sqrt(- x+5)-6
Graph:
Start by multiplying the output by 2 to find the function that represents the vertical stretch. Then, multiply the input by - 1 to reflect the graph in the y-axis. Finally, subtract 3 units from the input.
To help Ignacio, the transformations will be applied one at a time.
Distribute 2
Identity Property of Multiplication
To reflect the graph of a function in the y-axis, multiply the input by -1. Function g(x)=sqrt(x+2)-6 [1em] Reflection in they-axis y=g( -x) ⇕ y=sqrt(-x+2)-6 The graph of this function is a reflection in the y-axis of the graph of g. Let h(x) be this function. h(x)=sqrt(- x+2)-6
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graphs of the radical function y=asqrt(x-h)+k and its corresponding parent function y=sqrt(x) are given.
Start by considering a vertical stretch. Then, consider vertical and horizontal translations.
Disregard translations for a moment. In the graph of y=sqrt(x) it can be understood as, after moving 1 unit to the right of the starting point (0,0), the graph increases vertically 1 unit. Conversely, in the graph of y=asqrt(x-h)+k, after moving 1 unit to the right of the starting point (- 3,0), the graph increased vertically 2 units.
Identity Property of Addition
a-(- b)=a+b