H. Algebra 2
10.2 Notes
10.2 Graphing Square Root Functions
Date: ___________
Explore. Graphing and Analyzing the Parent Square Root Function
Although you have seen how to use imaginary numbers to evaluate the square roots of negative numbers, graphing
complex numbers and complex valued functions is beyond the scope of this course. For the purposes of graphing
functions based on the square roots (and in most cases where the square root function is used in a real-world example),
the domain and range should both be limited to real numbers.
The square root function is the inverse of a __________________________ function with a domain limited to 0 and
positive real numbers. The quadratic function must be a one-to-one function in order to have an inverse, so the domain
is limited to one side of the vertex. The square root function is also a one-to-one function as all inverse functions are.
Parent Square Root Function: 𝑓(𝑥) = √𝑥
A) Fill in the table and plot the points on the graph. Draw a smooth curve through the points to sketch the
graph. Then, give the graph’s endpoint and the domain and range of the function.
Endpoint: _____________________
Domain: _______________________ Range: ________________
B) The graph appears to be getting flatter as x increases, indicating that the rate of change
______________________ as x increases.
C) Describe the end behavior of the square root function, 𝑓(𝑥) = √𝑥.
Reflect.
1. Why does the end behavior of the square root function only need to be described at one end?
2. The solution to the equation 𝑥 2 = 4 is sometimes written as 𝑥 = ±2. Explain why the inverse of
𝑓(𝑥) = 𝑥 2 cannot similarly be written as 𝑔(𝑥) = ±√𝑥 in order to use all reals as the domain of f(x).
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Exploring Reflections of the Square Root Function
Graph each reflection of the parent function 𝒇(𝒙) = √𝒙 and give its domain and range.
g ( x)   x
Reflection across
_____________________
Domain: ______________
Range: ________________
h( x)   x
Reflection across
_____________________
i ( x)    x
Reflection across
_____________________
Domain: ______________
Range: ________________
Domain: ______________
Range: ________________
Graphing Square Root Functions
Learning Target E: I can graph square root functions of the form 𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘 or
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𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘 by hand.
Given each transformed square root function, tell the transformations that have been applied
to the parent graph. Then, find the endpoint and domain and range of the function to sketch
its graph.
A) 𝑔(𝑥) = 2√𝑥 − 3 − 2
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B) 𝑔(𝑥) = √− 2 (𝑥 − 2) + 1
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D) 𝑔(𝑥) = √3 (𝑥 + 2) + 1
C) 𝑔(𝑥) = −3√𝑥 − 2 + 3
Writing Square Root Functions
Learning Target F: I can write a square root function of the form 𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘 or
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𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘 given its graph.
Write the function that matches the graph using the given transformation format.
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A) 𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘
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B) 𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘
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C) 𝑔(𝑥) = √𝑏 (𝑥 − ℎ) + 𝑘
D) 𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘
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