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5. Rational Exponents & Radical Functions
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United States of AmericaTennessee
Algebra 2

5.03 Graphing radical functions

Lesson
Practice
Lesson

Concept summary

A radical function is a function that includes one or more radical expressions with the independent variable in the radicand.

The index of a radical function can be any real number but the most common are those containing square roots, called a square root function, and cube roots, called a cube root function. When dealing with square root functions, we need to consider the domain of the function before drawing a graph, as we cannot have negative values inside a square root if we want real numbers as the output.

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  • The graph of y=\sqrt{x}
  • Domain: \left[0, \infty \right)
  • Range: \left[0, \infty \right)
  • x-intercept: \left(0,0\right)
  • y-intercept: \left(0,0\right)
  • Increasing over its domain
  • As x \to \infty, y \to \infty
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  • The graph of y=\sqrt[3]{x}
  • Domain: \left(-\infty, \infty \right)
  • Range: \left(-\infty, \infty \right)
  • x-intercept: \left(0,0\right)
  • y-intercept: \left(0,0\right)
  • Point of inflection: \left(0,0\right)
  • Increasing over its domain
  • As x \to \infty, y \to \infty
  • As x \to -\infty, y \to -\infty
  • Odd function

Radical functions can be dilated, reflected, and translated in a similar way to other functions.

The square root parent function y=\sqrt{x} can be transformed to y=a\sqrt{x-h}+k

  • If a is negative, the basic curve is reflected across the x-axis
  • The graph is dilated vertically by a factor of a
  • The graph is translated to the right by h units
  • The graph is translated upwards by k units
  • The endpoint, originally at the origin, is moved to the point \left(h, k\right)
  • The domain of the graph becomes \left[h, \infty\right). Since the function is only defined when the term under the square root is non-negative, we need x-h\geq0 and so x\geq h.
  • The range of the graph becomes \left[k, \infty\right), for a>0, or \left(-\infty, k\right], for a<0, due to the vertical shift of k units

The cube root parent function y=\sqrt[3]{x} can be transformed to y=a\sqrt[3]{x-h}+k

Similar changes will apply to the cube root function, the only difference being the domain and range remain as all real x and y. The point of inflection will move to the point \left(h, k\right).

Worked examples

Example 1

Consider the following function:f\left(x\right) = \sqrt{x + 2}

a

Draw a graph of the function.

Approach

We will start by completing a table of values for the function, rounding to two decimal places where necessary:

x-2-1012347
f\left(x\right)011.411.7322.242.453

Solution

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b

Describe the transformation that occured to y=\sqrt{x} to give f\left(x\right).

Approach

The function is of the form f\left(x\right)=\sqrt{x-h} with h=-2.

Solution

The function has been translated to the left by 2 units.

c

State the domain and range of f\left(x\right).

Approach

As the graph has been translated 2 units to the left, the domain will change. The range will stay the same.

Solution

Domain: \left[-2, \infty\right)

Range: \left[0, \infty\right)

Example 2

Consider the following function:f \left( x \right) = - \sqrt[3]{x}+3

a

Draw a graph of the function.

Approach

We will start by completing a table of values for the function, rounding to two decimal places where necessary, and choosing some values such as 0, 1 and 8 that we know will evaluate to an integer value as they are perfect cubes.

x-8-2-101238
f\left(x\right)54.264321.741.561

Solution

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b

Describe the transformation that occured to y=\sqrt[3]{x} to give f\left(x\right).

Approach

The function is of the form f\left(x\right)=a\sqrt[3]{x}+k with a=-1, k=3.

Solution

The function has been reflected about the x-axis, and then translated up by 3 units.

c

State the domain and range of f\left(x\right).

Approach

The graph has been reflected across the x-axis and then translated 3 units up. As the function is a cube root function, the domain and range of the parent function was all real x and y. These translations do not change them.

Solution

Domain: \left(-\infty, \infty\right)

Range: \left(-\infty, \infty\right)

Example 3

Consider the graph of the function g\left(x\right)

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a

Identify if the function is increasing or decreasing over its domain.

Approach

The function is increasing if the y-values increase as x increases.

The function is decreasing if the y-values decrease as x increases.

Solution

We can see that the y-values are increasing as x increases, so the function is increasing over its domain.

b

Identify the transformation(s) that have occurred from f\left(x\right)=\sqrt{x}.

Approach

If we consider the graph of f\left(x\right)=\sqrt{x} compared to that of g\left(x\right)

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We can see that the starting point, or origin, of the graph has moved from \left(0,0\right) to \left(1, 2\right), so there has been both a horizontal and vertical shift. This point is not altered by a vertical compression or stretch as the starting values were both zero. The dashed line shows the result if only the two translations are applied.

We can see, however that the shape of the graph looks to be stretched vertically. This transformation is applied after the horizontal shift, but before the vertical shift.

We can see that the graph of g\left(x\right) increases 3 times faster than that of f\left(x\right), which means that g\left(x\right) has a vertical stretch by a factor of 3.

Solution

The parent function f\left(x\right)=\sqrt{x} has been translated 1 units to the right, stretched vertically by a factor of 3 and then translated 2 units upwards.

Reflection

Notice that we cannot find the vertical stretch factor by comparing f\left(x\right) with g\left(x\right) at the same x-values, because their starting points are different.

This is because the stretch happens before the vertical translation, but after the horizontal translation.

If we look at the general function: y=a\sqrt{x-h}+k we can see that \sqrt{x-h} is multiplied by the stretch factor a and then k is added.

c

State the equation of the function.

Approach

The general function is y=a\sqrt{x-h}+k, and we have

  • The graph is dilated vertically by a factor of a=3
  • The graph is translated to the right by h=1 units
  • The graph is translated upwards by k=2 units

Solution

The equation of the function is g\left(x\right)=3\sqrt{x-1}+2

Outcomes

A2.N.Q.A.1

Use units as a way to understand real-world problems.*

A2.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

A2.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

A2.F.IF.B.4

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP4

Model with mathematics.

A2.MP5

Use appropriate tools strategically.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

A2.MP8

Look for and express regularity in repeated reasoning.

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