Graph and analyze functions involving one square root

Alberta Grade 12 Mathematics

Relations and Functions

Alberta Mathematics 30-1 expectation RF13

The graph of the parent square-root function starts at one point and continues in one direction. Changes to the expression can move the endpoint, change which way the graph extends, or change its steepness. To analyze a square-root function, begin with the values allowed under the root. Then find the endpoint, range, and intercepts. These features give you a reliable sketch.

What you will learn

Find the domain and range of a function involving one square root.
Identify the endpoint and direction of a square-root graph.
Find intercepts and check that they are allowed.
Sketch a square-root graph using exact values when possible.

Review: square roots and allowed inputs

The principal square root of a non-negative number is its non-negative square root. For example, the principal square root of 9 is 3. The expression under a square-root symbol is called the radicand.

For a square root to have a real-number value, its radicand must be zero or positive. This restriction determines the domain, which is the set of allowed input values. The range is the set of output values the function can produce.

The parent function is the basic function used to recognize a family of related graphs. For square-root functions, it is the graph of the square root of x. It starts at the origin and continues right and up. Its domain and range both begin at zero.

y=\sqrt{x},\quad x\geq 0

A radicand cannot be negative when working with real outputs.
The domain lists allowed inputs; the range lists possible outputs.
The parent square-root graph begins at the origin.

Find the endpoint, domain, and range

A useful form is

y=a\sqrt{b(x-h)}+k

, where

a

and

b

are non-zero. The endpoint occurs where the radicand is zero. Substituting that input into the function gives the endpoint’s output.

To find the domain, require the radicand to be non-negative and solve the inequality. If

b

is positive, the graph extends from the endpoint toward larger x-values. If

b

is negative, it extends toward smaller x-values. The sign of

a

determines whether outputs rise or fall as the graph extends from the endpoint: positive

a

gives outputs at or above

k

, while negative

a

gives outputs at or below

k

.

The values of

a

and

b

also affect the graph’s steepness. A factor outside the root changes vertical steepness. A factor inside the root changes horizontal steepness. In particular, a factor greater than 1 multiplying the input inside the root makes the graph horizontally compressed compared with the corresponding graph without that factor.

For a sketch, start at the endpoint. Choose radicand values that are perfect squares, such as 0, 1, 4, and 9. Solve for the corresponding inputs and calculate the outputs. Plot these points and connect them with a smooth curve in the direction allowed by the domain.

b(x-h)\geq 0

Set the radicand equal to zero to find the endpoint input.
Solve the radicand inequality to determine which side of the endpoint is allowed.
The sign of the outside multiplier determines whether the graph rises or falls.

Find intercepts and check the graph

An x-intercept is a point where the graph has output zero. Set the whole function equal to zero to find one. Do not confuse the x-intercept with the endpoint: the endpoint is an x-intercept only when its output is zero.

A y-intercept is where the input is zero. Substitute zero for the input, but first make sure zero is in the domain. For either intercept, check that the result satisfies the original function and its domain restriction.

Keep exact values, such as a value containing a square root, when they are available. A decimal approximation can help describe or plot a value, but it does not replace an exact answer when one is requested.

As a final graph check, confirm that the endpoint, domain direction, range direction, and intercepts all match the equation. A few plotted points should also follow the curve from its endpoint.

y=0

Set the output to zero to find an x-intercept.
Substitute zero for the input to find a y-intercept, if that input is allowed.
Check intercepts in the original function and keep exact values where possible.

Practice before the check

For each function, find the domain, range, endpoint, direction of extension, and any intercepts. Sketch only after you have identified these features. Begin by requiring the radicand to be non-negative.

Practice A:

y=\sqrt{x+4}-2

. Practice B:

y=-\sqrt{2-x}+3

. Practice C:

y=\sqrt{2x+6}-1

. For each result, check that any proposed intercept lies in the domain and satisfies the function.

Solve the radicand inequality before sketching.
Use the endpoint and direction to guide the curve.
Substitute to verify intercepts.

Worked example

A graph shifted right and up

Analyze and sketch

y=\sqrt{x-2}+1

. State the domain, range, endpoint, and intercepts.

Shifted square-root graph

The graph shows a window of the full domain. It begins at its endpoint and continues to the right.

Find the allowed inputs
The expression under the square root must be non-negative. Solving that restriction gives all allowed inputs.
$x-2\geq 0\implies x\geq 2$
Locate the endpoint and range
The radicand is zero when the input is 2. Substitution gives the endpoint. The square root is non-negative, and adding 1 means the outputs begin at 1.
$(2,1),\quad y\geq 1$
Check the intercepts
The input 0 is outside the domain, so there is no y-intercept. To find an x-intercept, the square root would have to equal negative 1. A principal square root cannot be negative, so there is no x-intercept.

Answer: The domain is

x\geq 2

, and the range is

y\geq 1

. The endpoint is

(2,1)

. The graph extends right and upward. There are no x- or y-intercepts.

Check: At the endpoint input, the radicand is zero and the output is 1. The graph stays above the x-axis, as the range indicates.

Worked example

A graph extending left and downward

Analyze

y=-2\sqrt{3-x}+4

. State its domain, range, endpoint, and intercepts.

Square-root graph extending left

The graph shows a window of the full domain. It begins at its endpoint and continues to the left.

Determine the domain
The radicand must be non-negative. Solving the inequality shows that inputs cannot be greater than 3.
$3-x\geq 0\implies x\leq 3$
Find the endpoint and range
At input 3, the radicand is zero and the output is 4. The negative multiplier makes outputs decrease as the graph extends left from this endpoint.
$(3,4),\quad y\leq 4$
Find the intercepts
For the x-intercept, set the output equal to zero and solve. For the y-intercept, substitute zero, which is allowed by the domain.
$x=-1,\quad y(0)=4-2\sqrt{3}$

Answer: The domain is

x\leq 3

, and the range is

y\leq 4

. The endpoint is

(3,4)

. The x-intercept is

(-1,0)

, and the y-intercept is

(0,4-2\sqrt{3})

.

Check: At input

-1

, the radicand is 4, so the output is 0. At input 0, the exact output is

4-2\sqrt{3}

, which is approximately 0.54.

Worked example

A horizontal compression and an exact intercept

Analyze

y=\sqrt{2x+6}-1

. Find the domain, range, endpoint, and intercepts.

Horizontally compressed square-root graph

The graph shows a window of the full domain. It begins at x = −3 and continues to the right.

Find the domain
Require the radicand to be non-negative. Solving the inequality gives the full domain.
$2x+6\geq 0\implies x\geq -3$
Identify the endpoint and range
At the boundary input, the radicand is zero. The graph starts one unit below the x-axis and rises as the input increases. Rewriting the radicand as $2(x+3)$ shows a factor of 2 on the input relative to the parent graph, so the graph is horizontally compressed by a factor of one-half.
$(-3,-1),\quad y\geq -1$
Find both intercepts
For the x-intercept, set the output to zero and solve. For the y-intercept, substitute zero directly. Both inputs satisfy the domain.
$x=-\frac{5}{2},\quad y(0)=\sqrt{6}-1$

Answer: The domain is

x\geq -3

, and the range is

y\geq -1

. The endpoint is

(-3,-1)

. The graph extends right and upward and is horizontally compressed by a factor of one-half relative to the corresponding shifted parent graph. The x-intercept is

(-\frac{5}{2},0)

, and the y-intercept is

(0,\sqrt{6}-1)

.

Check: At input

-\frac{5}{2}

, the radicand is 1, so the output is 0. The y-intercept is exactly

\sqrt{6}-1

, approximately 1.45.

Common mistakes and how to avoid them

Allowing a negative radicand because the graph is reflected.

Correction: The radicand must still be non-negative. A reflection changes outputs, not the allowed inputs.

Calling every point where the radicand is zero an x-intercept.

Correction: That input gives the endpoint. It is an x-intercept only if the endpoint’s output is zero.

Assuming the domain always extends to the right.

Correction: Solve the radicand inequality. A negative coefficient on the input can make the domain extend left.

Treating a factor inside the root as a vertical stretch.

Correction: A factor multiplying the input inside the root changes the graph horizontally. A factor greater than 1 gives a horizontal compression.

Lesson summary

Require the radicand to be non-negative to find the domain.
Set the radicand equal to zero to find the endpoint input.
Use the outside multiplier and vertical shift to find the endpoint output and range.
Find intercepts by setting the appropriate coordinate to zero, then check the domain.
Sketch from the endpoint in the direction allowed by the domain.

Check your understanding

Question 1

For

y=\sqrt{5-x}+2

, which statement gives the domain and endpoint?

Domain $x\leq 5$ ; endpoint $(5,2)$
Domain $x\geq 5$ ; endpoint $(5,2)$
Domain $x\leq 5$ ; endpoint $(5,0)$
Domain all real numbers; endpoint $(0,2)$

Show answer and explanation

Domain

x\leq 5

; endpoint

(5,2)

The radicand restriction is

5-x\geq 0

, so

x\leq 5

. At

x=5

, the output is 2.

Question 2

What is the x-intercept of

y=\sqrt{x+1}-2

?

$(-1,0)$
$(3,0)$
$(0,3)$
There is no x-intercept.

Show answer and explanation

(3,0)

Set the output to zero. The square root must equal 2, so

x+1=4

and

x=3

. This input is in the domain.

Key terms

Radicand: The expression written under a square-root symbol.
Domain: The set of input values for which a function is defined.
Range: The set of output values a function can produce.
Endpoint: The starting point of a square-root graph.
Intercept: A point where a graph crosses an axis.
Parent function: The basic function used to recognize a family of related graphs.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF13. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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RF13 · Graph and analyze functions involving one square root

Alberta Grade 12 Mathematics

Alberta Mathematics 30-1 expectation RF13

What you will learn

Review: square roots and allowed inputs

Find the endpoint, domain, and range

Find intercepts and check the graph

Practice before the check

A graph shifted right and up

A graph extending left and downward

A horizontal compression and an exact intercept

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Math 30-1

About this lesson and its review