Square Root Functions

Definition and Graph of the Square Root Function

The square root of a nonnegative real number \( x \) is a number \( y \) such \( x = y^2 \). For example, \( 3 \) and \( - 3 \) are the square roots of \( 9 \) because \( 3^2 = 9 \) and \( (-3)^2 = 9 \)
Of the two roots, the nonnegative root of a nonnegative number \( x \) is written as function with the following notation \[ f(x) = \sqrt x\] where the symbol \( \sqrt { \; } \) is called the radical and \( x \) is called the radicand and must be nonnegative so that \( f(x) \) is real.

The square root function defined above is evaluated for some nonnegative values of \( x \) in the table below.

\( x \) \(f(x) = \sqrt x \)
\( 0 \) \(f(x) = \sqrt 0 = 0 \)
\( 1 \) \(f(x) = \sqrt 1 = 1\)
\( 2 \) \(f(x) = \sqrt 2 \approx 1.414\)
\( 4 \) \(f(x) = \sqrt 4 = 2\)
\( 7 \) \(f(x) = \sqrt 7 \approx 2.645\)
\( 9 \) \(f(x) = \sqrt 9 = 3 \)
\( 16 \) \(f(x) = \sqrt 16 = 4\)

The graph of the square root function is shown below with some points from the above table.

graph of square root function



Properties of the Square Root Function

Some of the properties of the square root function may be deduced from its graph

  1. The domain of the square root function \( f(x) = \sqrt x \) is given in interval form by: \( [0, + \infty) \)
  2. The range of the square root function \( f(x) = \sqrt x \) is given in interval form by: \( [0, + \infty) \)
  3. The x and y intercepts are both at \( (0,0) \)
  4. The square root function is an increasing function
  5. The square root function is a one-to-one function and has an inverse.



Common Mistakes to Avoid when Working with Square Root Functions

  1. It is wrong to write \( \sqrt{25} = \pm 5 \). The radicand is the symbol of the square root function and a function has only one output which as defined above is equal to the positive root.
    Correct answer: \( \sqrt{25} = 5 \)
  2. It is wrong to write \( \sqrt{x^2} = x \). The output of the square root is nonnenegative and \( x \) in the given expression may be negative, zero or postive.
    Correct answer: \( \sqrt{x^2} = |x| \)



Exploring Interactively the Square Root Function

Square root functions of the general form
\[ f(x) = a \sqrt{x - c} + d \] 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 \( d \) included in the expression of the square root function defined above. The Answers to the questions in the tutorial are included in this page.
click on the button above "draw" and start exploring.

a =
-10+10

c =
-10+10

d =
1+10

>


The answers to the following questions are included at the bottom of this page.
  1. 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.
  2. 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.
  3. 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.
  4. Use the sliders to set parameters \( a \), \( c \) and \( d \) to different values and determine which parameters affect the domain of the square root function \( f \) defined above? Find the domain analytically and compare it to the domain obtained graphically.
  5. Use the sliders to set parameters \( a \), \( c \) and \( d \) to different values and determine which parameters affect the range of the square root function \( f \) defined above? Find the range analytically and compare it to the range obtained graphically.
  6. How many x intercept the graph of \( f \) has?
  7. How many solutions an equation of the form
    \[ a \sqrt{x - c} + d = 0 \]
    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 graphically.
  8. Find the y intercept analytically and compare it to the one given by the app.

Answers to the Above Questions

  1. 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.
  2. 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.
  3. 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.
  4. Only parameter \( c \) affects the domain. The domain of
    \( f(x) = a \sqrt{x - c} + d = 0 \)
    may found by solving the inequality \( x - c \ge 0 \) hence the domain is the set of all values in the interval \( [c , + \infty) \)
  5. Only parameters \( a \) and \( d \) affect the range. The range of function \( f \) given above may found as follows: With x in the domain defined by interval \( [c , + \infty) \) , the \( \sqrt{x - c} \) is always positive or equal to zero hence
    \( \sqrt{x - c} \ge 0 \)
    If parameter \( a \) is positive then
    \( a \sqrt{x - c} \ge 0 \)
    Add \( d \) to both sides to obtain
    \( a \sqrt{x - c} + d \ge d \)
    Hence the range of the square root function defined above is the set of all values in the interval \( [d , + \infty) \)
    If parameter \( a \) is negative then
    \( a \sqrt{x - c} \le 0 \)
    Add \( d \) to both sides to obtain
    \( a \sqrt{x - c} + d \le d \)
    Hence the range of the square root function defined above is the set of all values in the interval \( (- \infty , d] \)
  6. one solution or no solution.
  7. Solve the equation
    \( a \sqrt{x - c} + d = 0\)

    add \( -d \) to both sides of the equation
    \( a \sqrt{x - c} = - d \)
    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.
  8. If \( c \) is positive, \( \sqrt{-c} \) 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
    \( y = a \sqrt{-c} + d \)

More References and Links to Square Root Functions

Square Root
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