Definition and Graph of the Square Root Function
The square root of a nonnegative real number \( x \) is a number \( y \) such that \( x = y^2 \). 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 a 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:
| \( x \) | \( f(x) = \sqrt{x} \) |
|---|---|
| \( 0 \) | \( \sqrt{0} = 0 \) |
| \( 1 \) | \( \sqrt{1} = 1 \) |
| \( 2 \) | \( \sqrt{2} \approx 1.414 \) |
| \( 4 \) | \( \sqrt{4} = 2 \) |
| \( 7 \) | \( \sqrt{7} \approx 2.645 \) |
| \( 9 \) | \( \sqrt{9} = 3 \) |
| \( 16 \) | \( \sqrt{16} = 4 \) |
The graph of the square root function is shown below with some points from the table above.
Properties of the Square Root Function
Some of the properties of the square root function may be deduced from its graph:
- The domain of the square root function \( f(x) = \sqrt{x} \) is given in interval form by: \( [0, +\infty) \)
- The range of the square root function \( f(x) = \sqrt{x} \) is given in interval form by: \( [0, +\infty) \)
- The x and y intercepts are both at \( (0,0) \)
- The square root function is an increasing function
- The square root function is a one-to-one function and has an inverse.
Common Mistakes to Avoid when Working with Square Root Functions
- It is wrong to write \( \sqrt{25} = \pm 5 \). The radical symbol denotes the principal square root function, and a function has only one output, which as defined above is equal to the positive root.
Correct answer: \( \sqrt{25} = 5 \) - It is wrong to write \( \sqrt{x^2} = x \). The output of the square root is nonnegative, and \( x \) in the given expression may be negative, zero, or positive.
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, and y-intercept are explored interactively. There is also another tutorial on graphing square root functions on 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.
Tutorial Questions and Answers
- Use the sliders to set parameters \( a \) and \( c \) to some constant values and change \( d \). What happens to 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 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 parameter \( a \). 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 root 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 root function \( f \) defined above? Find the range analytically and compare it to the range obtained graphically.
- How many x-intercepts does the graph of \( f \) have?
- How many solutions does an equation of the form \( a\sqrt{x-c} + d = 0 \) have? (\( a \neq 0 \)). Find the solution to this equation in terms of \( a \), \( c \), and \( d \) and compare it to the x-intercept given graphically.
- Find the y-intercept analytically and compare it to the one given by the app.
Answer
Changes in the parameter \( d \) affect the y-coordinates of all points on the graph, hence causing a vertical translation or shifting. When \( d \) increases, the graph is translated upward, and when \( d \) decreases, the graph is translated downward.
Answer
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.
Answer
Parameter \( a \) is a multiplicative factor for the y-coordinates of all points on the graph of function \( f \). If \( a > 1 \), the graph stretches (or expands) vertically. If \( 0 < a < 1 \), the graph shrinks vertically. If \( a \) changes sign, a reflection of the graph across the x-axis occurs.
Answer
Only parameter \( c \) affects the domain. The domain of \( f(x) = a\sqrt{x-c} + d \) may be found by solving the inequality \( x - c \ge 0 \), hence the domain is the set of all values in the interval \( [c, +\infty) \).
Answer
Only parameters \( a \) and \( d \) affect the range. With \( x \) in the domain \( [c, +\infty) \), \( \sqrt{x-c} \ge 0 \):
- If \( a > 0 \), then \( a\sqrt{x-c} \ge 0 \implies a\sqrt{x-c} + d \ge d \). Range: \( [d, +\infty) \).
- If \( a < 0 \), then \( a\sqrt{x-c} \le 0 \implies a\sqrt{x-c} + d \le d \). Range: \( (-\infty, d] \).
Answer
One solution or no solution (at most one x-intercept).
Answer
Solve the equation \( a\sqrt{x-c} + d = 0 \implies a\sqrt{x-c} = -d \implies \sqrt{x-c} = -\dfrac{d}{a} \).
If \( -d/a < 0 \), there is no solution. If \( -d/a \ge 0 \), squaring both sides gives \( x - c = \left(-\dfrac{d}{a}\right)^2 \implies x = \left(-\dfrac{d}{a}\right)^2 + c \).
Answer
If \( c > 0 \), \( \sqrt{-c} \) is not a real number, and therefore the graph has no y-intercept. If \( c \le 0 \), the y-intercept is given by evaluating \( f(0) = a\sqrt{-c} + d \).