2.06 Graphing square root functions
Consider the graph of y = \sqrt{ - x }:
As x approaches -\infty, what does y approach?
Consider the graph of y = \sqrt{x}:
Describe the rate of increase of the function as x increases.
State the axes intercepts.
Does the function have an asymptote?
Does the function have a limiting value?
As x increases, what does y approach?
Consider the function y = - \sqrt{x}.
Complete the table of values. Round any values to two decimal places if necessary.
Can the function values ever be positive?
| x | 0 | 1 | 2 | 3 | 4 | 5 | 9 |
|---|---|---|---|---|---|---|---|
| y |
The graph of the function y = - \sqrt{x} is shown. Is y = - \sqrt{x} an increasing function or a decreasing function?
Describe the rate of decrease of the function as x increases.
Consider the function y = \sqrt{ - x }.
Complete the table of values. Round any values to two decimal places if necessary.
| x | -5 | -4 | -3 | -2 | -1 | 0 |
|---|---|---|---|---|---|---|
| y |
The graph of y = \sqrt{ - x } is given.
Is y = \sqrt{ - x } an increasing function or a decreasing function?
Consider the function y = \sqrt{x} + 3.
Can y ever be negative?
As x gets larger and larger, what value does y approach?
Determine the y-intercept of the curve.
How many x-intercepts does it have?
Sketch the graph.
Consider the function y = 2 \sqrt{x} + 3.
Is the function increasing or decreasing from left to right?
Is the function more or less steep than y = \sqrt{x} ?
What are the coordinates of the vertex?
Sketch the graph.
Consider the function y = - \dfrac{1}{2} \sqrt{x} + 2.
Is the function increasing or decreasing from left to right?
Is the function more or less steep than y = \sqrt{x} ?
What are the coordinates of the vertex?
Sketch the graph.
Consider the function y = \sqrt{ - x }.
State the domain of the function.
State the range of the function.
Consider the function y = \sqrt{x}.
Complete the table of values. Round any values to two decimal places if necessary.
State the domain of the function.
State the range of the function.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 9 |
|---|---|---|---|---|---|---|---|
| y |
As x gets larger and larger, what value does y approach?
Sketch the graph of y = \sqrt{x}.
Consider the function y = - \sqrt{x}.
State the domain of the function.
State the range of the function.
The function y = \sqrt{x} has domain x \geq 0 and range y \geq 0.
What is the domain and range of y = \sqrt{x} - 2 ?
Consider the function y = \sqrt{x - 5}.
State the domain of the function.
State the range of the function.
Do the functions y = \sqrt{x} and y = \sqrt{x - 5} increase at the same rate?
Consider the function y = \sqrt{ - x } + 6.
What is the smallest possible function value?
State the domain of the function.
State the range of the function.
A square root function has a range of y \leq 0 and a domain of x \geq 0. Determine whether the following could be the equation of the function:
y = \sqrt{x}
y = - \sqrt{ - x }
y = - \sqrt{x}
y = 5 \sqrt{x}
y = - 5 \sqrt{x}
y = \sqrt{ - x }
For each of the following functions:
Sketch the graph
State the domain
State the range
f \left( x \right) = - \sqrt{x + 1}
f \left( x \right) = - 2 \sqrt{x + 5}
f \left( x \right) = 3 \sqrt{\left( \dfrac{1}{3} x\right)}
f \left( x \right) = - \dfrac{\sqrt{x}}{2} - 2
f \left( x \right) = \dfrac{\sqrt{x - 1}}{2} + 2
For each of the following functions:
State the domain of the function.
State the range of the function.
Sketch the graph.
For which values of x do the following expressions evaluate to a real number?
\sqrt{ 7 x}
\sqrt{x - 2}
\sqrt{3 - x}
\sqrt{15 - 5 x}
\sqrt{x^{2} + 6}
Consider the function f \left( x \right) = \sqrt{x - 2} + 3. State the domain of the function using interval notation.
The graph of y = \sqrt{x} has a vertex at \left(0, 0\right). By considering the transformations that have taken place, state the coordinates of the vertex of y = - \sqrt{x} + 3.
The graph of y = \sqrt{x} has been translated to the graph of y = \sqrt{x} - 4.
Describe the transformation that has occured on the original function.
Hence, sketch the graph of y = \sqrt{x} - 4.
Consider the graph of y = \sqrt{x} shown:
Sketch the curve after y = \sqrt{x} has been reflected about the y-axis.
What is the equation of this new graph?
Sketch the curve after y = \sqrt{x} has been reflected about the x-axis.
What is the equation of this new graph?
Consider the function y = \sqrt{x}:
Describe how we can transform the graph of y = \sqrt{x} to get the graph of y = \sqrt{x - 4} + 3.
Hence, sketch the graph of y = \sqrt{x - 4} + 3.
Sketch the curve y = 3 \sqrt{x - 2} + 3.