Key Concepts
- The features of the square root function.
- Identify the translation of the square root function.
- Calculate & compare the average rate of change of the square root functions.
Function, Domain, and Codomain
Function
A function is an equation for which any x that can be put into the equation will produce exactly one output such as out of the equation.
It is represented as y=fx , where is an independent variable and y is a dependent variable.
Example: y = x+2
Domain
A domain of a function is the set of inputs for which the function is defined.
Codomain
A codomain of a function is the set of possible output values.
Square root function
The function f(x) = √x is the square root function.
Graph of square root function:


Properties of square root function:
- Domain = All values of xx such that x≥0.
- Range = f(x)≥0.
- For f(x)=√x, the x–intercept and y-intercept of the graph of the function are both 0.
Note: The graph is increasing for all the values in the domain of f.
Example:
Sketch the graph p(x)=-√x and find the intercept, domain, and the range of the function.
Solution:


Function = √𝒇𝒙=−𝒙
Domain = All values of x such that x≥0.
Range = f(x)≤0 .
For f(x)=√x, the x–intercept and y–intercept of the graph of the function are both 0.
Translation of the square root function
Example: The graph of g(x)=√x+3 compared to the graph of f(x) = √x.
Solution:

The graph of g(x) = √x+3 is the vertical translation of f(x) = √x.
The domain for both the functions is x≥0.
The range for the function f is y≥0, so the range for the function g is y≥3.
Example:
The graph of g(x)=√x+3 compared to the graph of f(x)=√x.
Solution:

The graph of g(x)=√x+3 is the horizontal translation of f(x)=√x.
The domain for function f is x≥0, so the domain for the function g is x≥−3.
The range for both functions is y≥0.
Rate of change of square root function
Example:
For the function f(x)=√x , how does the average rate of change from x = 0 to x=0.3 compared to the average rate of change from x = 0.3 to x = 0.6?
Solution:

Step 1:
Evaluate the function for the x–values that correspond to the endpoints of each interval.
f(0) = √0 =0
f(0.3) = √0.3 ≈ 0.548
f(0.6) = √0.6 ≈ 0.775
Step 2:
Find the average rate of change over each interval.
From x=0 to x=0.3:
f(0.3)−f(0)/0.3−0 ≈ 0.548−0/0.3−0 = 0.548/0.3 ≈1.83
From x=0.3 to x=0.6:
f(0.6)−f(0.3)/0.6−0.3 ≈ 0.775−0.548/0.6−0.3 =0.227/0.3 ≈ 0.757
The average rate of change over the interval 0≤x≤0.3 is greater than the average rate of change over the interval 0.3≤x≤0.6.
Example:
Two plans are being considered to determine the speed of a theme park ride with a circular wall that spins. Plan A is represented by the function in the graph shown. The ride shown in the photo is an example of plan B. If the ride has a radius of 5 m, which plan will result in the greater speed for the ride?


Solution:
Plan A:
The graph of plan A shows that the corresponding speed at a radius of 4 m is about 6 m/s.
Plan B:
Evaluate f(r)=√2r for r=5.
f(5) = 2√5 ≈ 4.47
The ride using plan B has a speed of about 4.47 m/s when the radius is 5 m.
Conclusion:
With the radius of 4 m, the speed of the ride using plan A is 6 m/s.
The speed of the ride using plan B is about 4.47 m/s.
So, the ride using plan A has a greater speed for the radius of 4 m.
Exercise
- Compare the graph of g(x) = √x+4 to the graph of f(x) = √x .
- Compare the graph of p(x) = √x-2 to the graph of f(x) = √x .
- Find the average rate of change of f(x) = √3x ; 0≤x≤5 .
Concept Summary

What we have learned
- The features of square root function.
- The translation of the square root function from the graph.
- Calculate and compare the average rate of change of the square root functions.
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