Unlocking Graphing Radical Functions: A Comprehensive Guide

$x$	$2 4 6 - 3 x + 5$	$y$
$- 2$	$2 4 6 - 3 (- 2) + 5$	$\approx 8.72$
$- 1$	$2 4 6 - 3 (- 1) + 5$	$\approx 8.46$
$0$	$2 4 6 - 3 (0) + 5$	$\approx 8.13$
$1$	$2 4 6 - 3 (1) + 5$	$\approx 7.63$
$2$	$2 4 6 - 3 (2) + 5$	$5$

x

2 4 6 - 3 x + 5

y

- 2

2 4 6 - 3 (- 2) + 5

\approx 8.72

- 1

2 4 6 - 3 (- 1) + 5

\approx 8.46

0

2 4 6 - 3 (0) + 5

\approx 8.13

1

2 4 6 - 3 (1) + 5

\approx 7.63

2

2 4 6 - 3 (2) + 5

5

$y = a n b x + c + d,$ when $n$ is even
Sign of $a$	Range
Positive $(a > 0)$	$y \geq d$
Negative $(a < 0)$	$y \leq d$

y = a n b x + c + d,

when

n

is even

Sign of

a

Range

Positive

(a > 0)

y \geq d

Negative

(a < 0)

y \leq d

Since the index of the radical expression on the right-hand side of the equation is

2,

the given radical function is a square root function. Therefore, the radicand must be non-negative.

2 x \geq 0 \Leftrightarrow x \geq 0

This means that the domain of the function is the set of all real numbers greater than or equal to

0 .

To determine its range, let's compare the given function to the general form of a square root function.

General Form : Given Function : a b x + c + d \frac{1}{2} 2 x + 0 + 0

Since

a = \frac{1}{2}

is positive, the range of the function is all real numbers greater than or equal to

0 .

To determine the intercepts and end behavior, the function will be first drawn on a coordinate plane. To do so, a make a table of values. Be careful to use only

x -

values that belong to the domain in the table!

$x$	$\frac{1}{2} 2 x$	$y$
$0$	$\frac{1}{2} 2 (0)$	$0$
$1$	$\frac{1}{2} 2 (1)$	$\approx 0.71$
$2$	$\frac{1}{2} 2 (2)$	$1$
$3$	$\frac{1}{2} 2 (3)$	$\approx 1.22$
$4$	$\frac{1}{2} 2 (4)$	$\approx 1.41$

Next, the obtained points can be plotted and connected with a smooth curve.

From the graph, it is seen that the $x -$ intercept and the $y -$ intercept both occur at the origin. It can also be seen that this function increases over its entire domain and that $y$ tends to infinity as $x$ tends to infinity. With this information, the desired characteristics can be written. Recall that the domain and range are both all non-negative real numbers!

Domain	$x \geq 0$
Range	$y \geq 0$
$x -$ intercept	$x = 0$
$y -$ intercept	$y = 0$
End Behavior	$y x \to 0 ⟶ 0$ and $y x \to + \infty ⟶ + \infty$

$x$	$5 3 x + 1 - 2$	$y$
$- 3$	$5 3 (- 3) + 1 - 2$	$\approx - 3.52$
$- 2$	$5 3 (- 2) + 1 - 2$	$\approx - 3.38$
$- 1$	$5 3 (- 1) + 1 - 2$	$\approx - 3.15$
$0$	$5 3 (0) + 1 - 2$	$- 1$
$1$	$5 3 (1) + 1 - 2$	$\approx - 0.68$
$2$	$5 3 (2) + 1 - 2$	$\approx - 0.52$
$3$	$5 3 (3) + 1 - 2$	$\approx - 0.42$

x

5 3 x + 1 - 2

y

- 3

5 3 (- 3) + 1 - 2

\approx - 3.52

- 2

5 3 (- 2) + 1 - 2

\approx - 3.38

- 1

5 3 (- 1) + 1 - 2

\approx - 3.15

0

5 3 (0) + 1 - 2

- 1

1

5 3 (1) + 1 - 2

\approx - 0.68

2

5 3 (2) + 1 - 2

\approx - 0.52

3

5 3 (3) + 1 - 2

\approx - 0.42

$x$	$3 x + 1$	$y$
$- 4$	$3 - 4 + 1$	$\approx - 1.44$
$- 3$	$3 - 3 + 1$	$\approx - 1.26$
$- 2$	$3 - 2 + 1$	$- 1$
$- 1$	$3 - 1 + 1$	$0$
$0$	$3 0 + 1$	$1$
$1$	$3 1 + 1$	$\approx 1.26$
$2$	$3 2 + 1$	$\approx 1.44$
$3$	$3 3 + 1$	$\approx 1.59$
$4$	$3 4 + 1$	$\approx 1.71$

x

3 x + 1

y

- 4

3 - 4 + 1

\approx - 1.44

- 3

3 - 3 + 1

\approx - 1.26

- 2

3 - 2 + 1

- 1

- 1

3 - 1 + 1

0

0

3 0 + 1

1

1

3 1 + 1

\approx 1.26

2

3 2 + 1

\approx 1.44

3

3 3 + 1

\approx 1.59

4

3 4 + 1

\approx 1.71

Domain	All real numbers
Range	All real numbers
$x -$ intercept	$x = - 1$
$y -$ intercept	$y = 1$
End Behavior	$y x \to - \infty ⟶ - \infty$ and $y x \to + \infty ⟶ + \infty$

Domain

All real numbers

Range

All real numbers

x -

intercept

x = - 1

y -

intercept

y = 1

End Behavior

y x \to - \infty ⟶ - \infty

and

y x \to + \infty ⟶ + \infty

$x$	$3 2 x - 4 + 3$	$y$
$- 4$	$3 2 (- 4) - 4 + 3$	$\approx 0.71$
$- 3$	$3 2 (- 3) - 4 + 3$	$\approx 0.85$
$- 2$	$3 2 (- 2) - 4 + 3$	$1$
$- 1$	$3 2 (- 1) - 4 + 3$	$\approx 1.18$
$0$	$3 2 (0) - 4 + 3$	$\approx 1.41$
$1$	$3 2 (1) - 4 + 3$	$\approx 1.74$
$2$	$3 2 (2) - 4 + 3$	$3$
$3$	$3 2 (3) - 4 + 3$	$\approx 4.26$
$4$	$3 2 (4) - 4 + 3$	$\approx 4.59$

x

3 2 x - 4 + 3

y

- 4

3 2 (- 4) - 4 + 3

\approx 0.71

- 3

3 2 (- 3) - 4 + 3

\approx 0.85

- 2

3 2 (- 2) - 4 + 3

1

- 1

3 2 (- 1) - 4 + 3

\approx 1.18

0

3 2 (0) - 4 + 3

\approx 1.41

1

3 2 (1) - 4 + 3

\approx 1.74

2

3 2 (2) - 4 + 3

3

3

3 2 (3) - 4 + 3

\approx 4.26

4

3 2 (4) - 4 + 3

\approx 4.59

$x$	$3 3 x - 6 - 1$	$y$
$- 3$	$3 3 (- 3) - 6 - 1$	$\approx - 3.47$
$- 2$	$3 3 (- 2) - 6 - 1$	$\approx - 3.29$
$- 1$	$3 3 (- 1) - 6 - 1$	$\approx - 3.08$
$0$	$3 3 (0) - 6 - 1$	$\approx 2.82$
$1$	$3 3 (1) - 6 - 1$	$\approx - 2.44$
$2$	$3 3 (2) - 6 - 1$	$- 1$
$3$	$3 3 (3) - 6$

x

3 3 x - 6 - 1

y

- 3

3 3 (- 3) - 6 - 1

\approx - 3.47

- 2

3 3 (- 2) - 6 - 1

\approx - 3.29

- 1

3 3 (- 1) - 6 - 1

\approx - 3.08

0

3 3 (0) - 6 - 1

\approx 2.82

1

3 3 (1) - 6 - 1

\approx - 2.44

2

3 3 (2) - 6 - 1

- 1

3

3 3 (3) - 6

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Catch-Up and Review

Reflecting the Graph of a Quadratic Function

Radical Functions and Square Root Functions

Radical Function

Square Root Function

Domain and Range of Even Indexed Radical Functions

Football and Square Root Functions

Answer

Hint

Solution

Cube Root Functions

Domain and Range of Odd Indexed Radical Functions

Football and Cube Root Functions

Answer

Hint

Solution

Finding the Domain and Range of Radical Functions

Radical Inequalities

Graphing a Radical Inequality in Two Variables

Football and Cube Root Inequalities

Answer

Hint

Solution

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