Square root
functions of the form
f(x) = a √(x  c) + d, parameter a not equal to zero
and the characteristics of their graphs such as domain, range, x intercept, y intercept are explored interactively. Square root equations are also explored graphically. There is also another tutorial on
graphing square root functions in this site.
The exploration is carried out by changing the parameters a, c, and d defining the square root function defined above. Answers to the questions in the tutorial are included in this page.
Tutorial
click on the button above "draw" and start exploring.
The answers to the following questions are included in this page.
 Use the sliders to set parameters a and c to some constant values and change d. What happens to the the graph when the value of parameter d changes? Give an analytical explanation.
 Use the sliders to set parameters a and d to some constant values and change c. What happens to the the graph when the value of parameter c changes? Give an analytical explanation.
 Use the sliders to set parameters c and d to some constant values and change parameters a. What happens to the graph when the value of parameter a changes? Give an analytical explanation.
 Use the sliders to set parameters a, c and d to different values and determine which parameters affect the domain of the square function f defined above? Find the domain analytically and compare it to the domain obtained graphically.
 Use the sliders to set parameters a, c and d to different values and determine which parameters affect the range of the square function f defined above? Find the range analytically and compare it to the range obtained graphically.
 How many x intercept the graph of f has?

How many solutions an equation of the form
a √(x  c) + d = 0
has? (parameter a not equal to zero). Find the solution to this equation in terms of a, c and d and compare it to the x intercept given by the applet.
 Find the y intercept analytically and compare it to the one given by the applet.
Answers to the Above Questions

Changes in the parameter d affect the y coordinates of all points on the graph hence the vertical translation or shifting. When d increases, the graph is translated upward and when d decreases the graph is translated downward.

When c increases, the graph is translated to the right and when c decreases, the graph is translated to the left. This is also called horizontal shifting.

Parameter a is a multiplicative factor for the y coordinates of all points on the graph of function f. Let a be greater than zero. If a gets larger than 1, the graph stretches (or expands) vertically. If a gets smaller than 1, the graph shrinks vertically. If a changes sign, a reflection of the graph on the x axis occurs.
 Only parameter c affects the domain. The domain of
f(x) = a √(x  c) + d = 0
may found by solving the inequality x  c >= 0 hence the domain is the set of all values in the interval [c , + ∞).

Only parameters a and d affect the range. The range of f given above may found as follows: With x in the domain defined by interval [c , + ∞), the √(x  c) is always positive or equal to zero hence
√(x  c) >= 0
If parameter a is positive then
a √(x  c) >= 0
Add d to both sides to obtain
a √(x  c) + d >= d
Hence the range of the square root function defined above is the set of all values in the interval [d , + ∞)
If parameter a is negative then
a √(x  c) <= 0
Add d to both sides to obtain
a √(x  c) + d <= d
Hence the range of the square root function defined above is the set of all values in the interval ( ∞ , d]
 one solution or no solution.

Solve the equation
a √(x  c) + d = 0
add d to both sides of the equation
a √(x  c) =  d
If d is positive  d is negative and the above equation has no solution. If d is negative, we square both sides and solve to obtain
x = (d/a)^{2} + c
The above equation may have one solution or no solution.
 If c is positive, √(c) is not a real number and therefore the graph has no y intercept. If c is negative or equal to zero then the y intercept is given by
y = a √(c) + d
More References and Links to Graphingfunctions.
