Radical functions & their graphs

Google Classroom

Practice some problems before going into the exercise.

Introduction

In this article, we will practice a couple of problems where we should match the appropriate graph to a given radical function.

Practice question 1: Square-root function

The graph of

y = \sqrt{x}

is shown below.

A square root function graph on an x y coordinate plane. It has an endpoint (zero, zero) and passes through (four, two).

Which of the following is the graph of $y = - \sqrt{x + 3} - 5$ ‍?

The graph of

y = \sqrt{x}

can be transformed to get the graph of

y = - \sqrt{x + 3} - 5

Replacing $x$ ‍ with $x + 3$ ‍ shifts the graph of $y = \sqrt{x}$ ‍ to the left $3$ ‍ units.

A square root function graph on an x y coordinate plane and its shifted graph. It has an endpoint (zero, zero) and passes through (four, two). The shifted graph has an endpoint at (negative three, zero) and passes through (one, two).

Multiplying $\sqrt{x + 3}$ ‍ by $- 1$ ‍ reflects the graph of $y = \sqrt{x + 3}$ ‍ over the $x$ ‍-axis.

A square root function graph on an x y coordinate plane and its reflected graph across the x-axis. It has an endpoint at (negative three, zero) and passes through (one, two). The reflected graph has an endpoint at (negative three, zero) and passes through (one, negative two).

Adding $- 5$ ‍ to the function shifts the graph of $y = - \sqrt{x + 3}$ ‍ down $5$ ‍ units.

A square root function graph on an x y coordinate plane and its shifted graph. It has an endpoint at (negative three, zero) and passes through (one, negative two). The shifted graph as an endpoint at (negative three, negative five) and passes through (one, negative seven).

Note that it is important to follow order of operations when building the function, as function transformations are not always commutative.

In conclusion, the graph that corresponds with these transformations is graph $A$ ‍:

A square root function graph on an x y coordinate plane. It has an endpoint (negative three, negative five) and passes through (six, negative eight).

Practice question 2: Cube-root function

The graph of

y = \sqrt[3]{x}

is shown below.

A cube root function graph on an x y coordinate plane. Its middle point is at (zero, zero). It passes through (negative eight, negative two) and (eight, two).

Which of the following is the graph of $y = - \sqrt[3]{x + 2} + 5$ ‍?

The graph of

y = \sqrt[3]{x}

can be transformed to get the graph of

y = - \sqrt[3]{x + 2} + 5

Replacing $x$ ‍ with $x + 2$ ‍ shifts the graph of $y = \sqrt[3]{x}$ ‍ left $2$ ‍ units.

A cube root function graph and its shifted graph on an x y coordinate plane. Its middle point is at (zero, zero). It passes through (negative eight, negative two) and (eight, two). The shifted graph has a middle point is at (negative two, zero). It passes through (negative ten, negative two) and (six, two).

Multiplying $\sqrt[3]{x + 2}$ ‍ by $- 1$ ‍ reflects the graph of $y = \sqrt[3]{x + 2}$ ‍ over the $x$ ‍-axis.

A cube root function graph and its horizontally reflected graph on an x y coordinate plane. Its middle point is at (negative two, zero). It passes through (negative ten, negative two) and (six, two). The reflected graph has its middle point at (negative two, zero). It passes through (negative ten, two) and (six, negative two).

Adding $5$ ‍ to the function shifts the graph of $y = - \sqrt[3]{x + 2}$ ‍ up $5$ ‍ units.

A cube root function graph and its shifted graph on an x y coordinate plane. Its middle point is at (negative two, zero). It passes through (negative ten, two) and (six, negative two). The shifted graph has its middle point at (negative two, five). It passes through (negative ten, seven) and (six, three).

Note that it is important to follow order of operations when building the function, as function transformations are not always commutative.

In conclusion, the graph that corresponds with these transformations is graph $C$ ‍:

A cube root function graph on an x y coordinate plane. Its middle point is at (negative two, five). It passes through (negative ten, seven) and (six, three).

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