Evaluating and graphing functions that contain a square root. Includes the square root function and its domain and range, finding the domain by requiring the radicand to be non-negative, building a table of values and sketching the curve, the effect of a constant outside the radical on the range, and the effect of a constant inside it on the domain.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 12 — Radicals and More Connections to Geometry
Functions Involving Square Roots
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-697 — the lesson these objectives are drawn from
Warm-up
Lesson 9.1 established that a negative number has no real square root. That single fact decides the domain of every function in this lesson.
Discussion prompt
For which values of x is the square root of x a real number? What does that say about where its graph can be drawn?
Hint: Negative radicands are undefined.
Answer:
\[ \sqrt{x} \text{ is defined exactly when } x \ge 0 \]
So the graph exists only to the right of the vertical axis and, since square roots are never negative, only above the horizontal one. It occupies a single quarter of the plane, which is quite unlike anything earlier in the course.
Concept
The square root function is y equals the square root of x. Its domain is all non-negative numbers and its range is all non-negative numbers, because a radicand may not be negative and a square root is never negative.
square root function — The function y equals the square root of x, whose domain and range are both the set of non-negative real numbers.
Other functions involving square roots behave similarly.
Figure (svg): The graph of the square root function
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-692
Section
Section 1
Concept
A square root is defined only when the expression under it is non-negative. Setting the radicand greater than or equal to nought and solving gives the domain.
\[ \text{radicand} \ge 0 \]
Find the domain before making a table.
Figure (svg): The domain and range of a square root function marked on the axes
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-692 — Example 1 and its Study Tip on when a square root can be evaluated
Picture it
Inputs across, outputs up.
Figure (svg): The domain and range of a square root function marked on the axes
Knowing the domain first tells you which values of x are worth putting in a table. Choosing values outside it wastes effort on entries that do not exist.
Worked example
This is Example 1 from the textbook.
\[ \text{Find the domain of } y = 2\sqrt{x} \text{ and tabulate some values.} \]
Set the radicand non-negative
Why: The radicand is just x.
\[ x \ge 0 \]
State the domain
Why: All non-negative numbers.
\[ x \ge 0 \]
Choose values in it
Why: Nought to five.
\[ 0, 1, 2, 3, 4, 5 \]
Compute the outputs
Why: Twice each square root.
\[ 0, 2, 2.8, 3.5, 4, 4.5 \]
Figure (svg): The domain and range of a square root function marked on the axes
\[ \text{domain: } x \ge 0 \]
Verify: check two entries
Why: At x equal to four the root is two and twice it is four. At x equal to two the root is about 1.414 and twice it is about 2.83, which rounds to 2.8. The perfect squares give exact values and the others need rounding.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-692
Faded example
Then solve for x.
Fill in the blanks
y = \sqrt0: \quad 2x \ge 0 \;\Longrightarrow\; x \ge ___
Why: Dividing a non-negative quantity by two leaves it non-negative, so the domain is unchanged. A positive coefficient inside the radical never moves the domain.
Worked example
Guided Practice 1 to 4, with different radicands.
\[ \text{Find the domain of } y = \sqrt{x}, \; y = 3\sqrt{x}, \; y = \sqrt{2x} \text{ and } y = \sqrt{x} + 1. \]
Take the first two
Why: The radicand is x in both.
\[ x \ge 0 \]
Take the third
Why: The radicand is two x.
\[ 2x \ge 0 \]
Solve it
Why: Divide by two.
\[ x \ge 0 \]
Take the fourth
Why: The one is outside the radical.
\[ x \ge 0 \]
Figure (svg): Two columns separating changes inside the radical from changes outside it
\[ x \ge 0 \text{ in every case} \]
Verify: notice what did not matter
Why: The multiplier three, the coefficient two inside, and the plus one outside all left the domain unchanged. Only a constant added or subtracted inside the radical moves it, which is the subject of a later section.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-692
Trap
\[ y = \sqrt{x - 3} \;\Longrightarrow\; \text{domain } x \ge 0 \]
Copy the domain of the basic square root function
Why: Every example so far had domain x at least nought.
The radicand here is x minus three, not x, so the condition is x minus three at least nought. That gives x at least three, and values between nought and three are not in the domain at all.
\[ x - 3 \ge 0 \;\Longrightarrow\; x \ge 3 \]
Write the radicand's inequality and solve it
Why: Whatever the radicand happens to be.
Testing one value inside and one outside confirms it in a few seconds.
Sorting
For y equals the square root of x minus 3.
Sort into buckets
Sort each value by whether the function is defined there.
The boundary value of three is included, because the square root of nought is nought — a perfectly good output. It is only strictly negative radicands that are excluded.
Elimination
For a function containing a square root.
Eliminate the wrong options
What must be non-negative?
Survives elimination: A
Why: The restriction comes from the square root itself, so it applies to whatever sits underneath it. Option B is right in the simplest cases and wrong as soon as the radicand is anything but x.
Socratic
The table could be built first.
Discussion prompt
Say what finding the domain first tells you about the table. Then say what happens if you choose values outside it.
Hint: Which x values will produce outputs?
Answer:
It tells you which values of x are worth using, so every entry in the table will actually exist. It also tells you where the curve begins, which is the single most informative point on the graph.
Choosing values outside the domain produces entries with no value at all — a calculator returns an error and there is nothing to plot. Worse, a careless calculation might produce a number anyway and put a point on the graph where the function has none, which misrepresents the whole picture.
Section
Section 2
Concept
Once the domain is known, choose values within it, compute the outputs, plot the points and join them with a smooth curve. The range is the set of outputs the curve reaches.
Perfect squares make the tidiest table entries.
Figure (svg): A square root function stretched by a factor
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693 — Example 2, Graph y equals two root x, and its Study Tip on choosing several values
Picture it
Same domain, doubled outputs.
Figure (svg): A square root function stretched by a factor
The multiplier changes how steeply the curve rises but not where it starts. Both graphs begin at the origin and both accept the same inputs.
Worked example
This is Example 2 from the textbook.
\[ \text{Sketch } y = 2\sqrt{x} \text{ and state its range.} \]
Use the domain
Why: All non-negative numbers.
\[ x \ge 0 \]
Plot the tidy points
Why: Where x is a perfect square.
\[ (0, 0), (1, 2), (4, 4) \]
Join smoothly
Why: A curve, not segments.
Read the range
Why: Outputs from nought upwards.
\[ y \ge 0 \]
Figure (svg): A square root function stretched by a factor
\[ \text{range: } y \ge 0 \]
Verify: check the shape against the table
Why: Between x equal to one and four the output rose from two to four, and between four and nine it would rise only from four to six. Equal steps in x give ever smaller steps in y, which is the flattening the curve shows.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693
Faded example
From the lowest output upwards.
Fill in the blanks
For y equals 2 root x, the smallest output is 0, so the range is all y greater than or equal to 0.
Why: The smallest input gives the smallest output because the function only rises. The range therefore starts where the curve does and extends upwards without bound.
Worked example
Making the table easy on purpose.
\[ \text{Which values of } x \text{ give exact outputs for } y = 2\sqrt{x}? \]
Look for perfect squares
Why: Their roots are whole numbers.
\[ 0, 1, 4, 9 \]
Compute those outputs
Why: Twice each root.
\[ 0, 2, 4, 6 \]
Note the others
Why: Two, three and five need rounding.
\[ 2.8, 3.5, 4.5 \]
Decide what to plot
Why: Exact points, plus a few between.
Figure (svg): A square root function stretched by a factor
\[ x = 0, 1, 4, 9 \;\Longrightarrow\; y = 0, 2, 4, 6 \]
Verify: see why this helps
Why: Plotting four exact points is faster and more accurate than plotting six rounded ones, and the shape between them is smooth enough to sketch confidently. Choosing perfect squares deliberately is worth doing every time.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693
Error analysis
The student graphed the square root function.
Annotate
On: \( \begin{aligned} y &= \sqrt{x} \\ \text{at } x = 4: \quad y &= \pm 2 \\ \text{so the graph has two branches} \end{aligned} \)
This confuses solving an equation, where both roots are wanted, with evaluating a function, where the radical names one value. Lesson 9.1 drew that distinction and it matters here for the shape of the entire graph.
Prediction
As x increases steadily.
Predict first
How do the outputs change?
Correct: They rise, but by less and less.
\[ \sqrt{1} = 1, \; \sqrt{4} = 2, \; \sqrt{9} = 3, \; \sqrt{16} = 4 \]
Why: Going from x equal to one to four raises the output of two root x from two to four, but going from four to nine raises it only from four to six — a bigger step in x for a smaller step in y. The curve therefore flattens as it goes right, though it never stops rising and never reaches a ceiling. That is the opposite behaviour to the exponential growth of Chapter 8.
Elimination
For a square root function.
Eliminate the wrong options
Which inputs give exact outputs?
Survives elimination: A
Why: Perfect squares give whole-number roots, which land exactly on gridlines and make a hand-drawn graph both quicker and more accurate. Choosing them deliberately is a small habit worth having.
Socratic
Every positive number has two square roots.
Discussion prompt
Explain why the graph does not have a second branch below the axis. Then say what curve you would get if it did.
Hint: How many outputs may a function have per input?
Answer:
The radical symbol names the positive square root only, so each input produces exactly one output — which is what a function requires. A second branch would mean two outputs for the same input, and the graph would fail the test for being a function at all.
Both branches together would form a curve opening sideways, which is a parabola turned on its side. That is a perfectly good curve but not a function of x, which is why the square root function takes only half of it — and why the plus-or-minus has to be written explicitly when solving equations.
Section
Section 3
Concept
Adding a constant outside the radical raises every output by that amount. The domain is unaffected, because the radicand has not changed.
\[ y = \sqrt{x} + 1 \]
The range starts at the constant instead of at nought.
Figure (svg): A square root graph raised by adding a constant
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693 — Example 3, Graph y equals root x plus one
Picture it
Same shape, one unit higher.
Figure (svg): A square root graph raised by adding a constant
Every point moved straight up by one, so the curve begins at nought, one rather than at the origin. Its shape and its domain are exactly as before.
Worked example
This is Example 3 from the textbook.
\[ \text{Find the domain of } y = \sqrt{x} + 1, \text{ sketch it and find the range.} \]
Find the domain
Why: The radicand is still x.
\[ x \ge 0 \]
Build the table
Why: Add one to each root.
\[ 1, 2, 2.4, 2.7, 3, 3.2 \]
Plot and join
Why: The same shape, lifted.
\[ \text{starts at } (0, 1) \]
Read the range
Why: From one upwards.
\[ y \ge 1 \]
Figure (svg): A square root graph raised by adding a constant
\[ \text{domain } x \ge 0, \quad \text{range } y \ge 1 \]
Verify: check two table entries
Why: At x equal to four the root is two and adding one gives three. At x equal to two the root is about 1.414 and adding one gives about 2.41, which rounds to 2.4. Both match the table.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693
Faded example
The domain is untouched.
Fill in the blanks
For y equals root x plus 1, the domain is x at least 0 and the range is y at least 1.
Why: The plus one lifted every output by one, so the smallest output is one rather than nought. The set of permitted inputs did not change at all.
Worked example
Guided Practice 9, where the curve is reflected and lowered.
\[ \text{Find the domain and range of } y = 3 - \sqrt{x}. \]
Find the domain
Why: The radicand is x.
\[ x \ge 0 \]
Find the largest output
Why: At x equal to nought.
\[ y = 3 \]
See what happens as x grows
Why: The root grows, so y falls.
State the range
Why: Three and everything below.
\[ y \le 3 \]
Figure (svg): A square root graph raised by adding a constant
\[ \text{domain } x \ge 0, \quad \text{range } y \le 3 \]
Verify: test a large input
Why: At x equal to a hundred the root is ten, so y is negative seven. The outputs fall without bound, so the range extends downwards forever from three — which is the reverse of the usual direction.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 693-693
Trap
\[ y = \sqrt{x} + 1 \;\Longrightarrow\; x + 1 \ge 0 \;\Longrightarrow\; x \ge -1 \]
Include the plus one in the radicand's inequality
Why: It is part of the expression, so it went into the condition.
The one is outside the radical and has nothing to do with what may go under it. At x equal to negative one half the square root of that value is still undefined, so the domain really does start at nought.
\[ x \ge 0 \]
Write out only what is under the radical sign, then solve
Why: Position decides everything here.
Reading the expression carefully once is quicker than testing a value to discover the error.
Matching
Look at the constant outside.
Match the pairs
Why: The first three all start at their constant and rise, and the fourth starts at three and falls because the root is subtracted. The direction of the range depends on the sign in front of the radical.
Prediction
As in y equals 3 minus the root of x.
Predict first
How does the graph behave?
Correct: It starts at 3 and falls, ever more slowly.
\[ x = 0: \; y = 3 \qquad x = 4: \; y = 1 \qquad x = 9: \; y = 0 \]
Why: At x equal to nought the root is nought, so y is three; as x grows the root grows and is subtracted, so y decreases. The flattening remains, since the root itself flattens — the outputs fall quickly at first and then more gently. The graph is the ordinary square root curve turned upside down and lifted to three.
Socratic
It is part of the same expression.
Discussion prompt
Explain why only the radicand determines the domain. Then say what an outside constant does determine.
Hint: Where does the restriction come from?
Answer:
The restriction comes from the square root operation itself, which refuses negative inputs. A constant added afterwards is applied to a number that has already been produced, so it cannot make an undefined value defined or the reverse — by then the damage, or the success, has already happened.
What it determines is the range, since every output is shifted by that amount. Domain and range are therefore controlled by different parts of the expression, which is why they must be found separately rather than read off together.
Section
Section 4
Concept
When a constant is subtracted inside the radical, the radicand's inequality changes and so does the domain. The curve starts wherever the radicand first reaches nought.
\[ y = \sqrt{x - 3} \;\Longrightarrow\; x - 3 \ge 0 \]
Solve the inequality to find where the graph begins.
Figure (svg): A square root graph whose domain has been shifted
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 694-694 — Example 4, Graph y equals root of x minus three
Picture it
Nothing before x equals three.
Figure (svg): A square root graph whose domain has been shifted
The shape is identical to the basic square root curve, moved three units to the right. Its range is unchanged, since the outputs still begin at nought.
Worked example
This is Example 4 from the textbook.
\[ \text{Find the domain of } y = \sqrt{x - 3} \text{ and sketch it.} \]
Write the inequality
Why: The radicand is non-negative.
\[ x - 3 \ge 0 \]
Solve it
Why: Add three to each side.
\[ x \ge 3 \]
Choose values in the domain
Why: Starting at three.
\[ 3, 4, 7, 12 \]
Compute and plot
Why: Roots of nought, one, four and nine.
\[ 0, 1, 2, 3 \]
Figure (svg): A square root graph whose domain has been shifted
\[ \text{domain: } x \ge 3 \]
Verify: test either side of three
Why: At x equal to four the radicand is one and the output is one. At x equal to two the radicand is negative one, which has no real square root — so the function genuinely stops at three rather than merely being small there.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 694-694
Faded example
One line of algebra.
Fill in the blanks
x - 3 \ge 0 \;\Longrightarrow\; x \ge 3, \quad \text3 x = ___
Why: The curve begins exactly where the radicand first reaches nought, which is the smallest permitted input. That point is always on the horizontal axis when nothing is added outside.
Worked example
Making the radicand a perfect square.
\[ \text{Which values of } x \text{ give exact outputs for } y = \sqrt{x - 3}? \]
Ask what the radicand should be
Why: A perfect square.
\[ 0, 1, 4, 9 \]
Add three to each
Why: To recover x.
\[ 3, 4, 7, 12 \]
Compute the outputs
Why: The roots themselves.
\[ 0, 1, 2, 3 \]
Note the pattern
Why: Inputs spaced ever further apart.
\[ 1, 3, 5\text{ apart} \]
Figure (svg): A square root graph whose domain has been shifted
\[ x = 3, 4, 7, 12 \;\Longrightarrow\; y = 0, 1, 2, 3 \]
Verify: check the spacing
Why: The gaps between the chosen inputs are one, three and five — the odd numbers, because consecutive squares differ by consecutive odd numbers. That spreading is exactly why the curve flattens.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 694-694
Trap
\[ y = \sqrt{x - 3} \;\Longrightarrow\; \text{domain } x \ge -3 \]
Move the three across with its sign unchanged
Why: The three was negative, so the domain seemed to start at negative three.
Solving x minus three at least nought means adding three, giving x at least three. Testing x equal to nought settles it: the radicand would be negative three, which has no real root.
\[ x - 3 \ge 0 \;\Longrightarrow\; x \ge 3 \]
Solve the inequality properly rather than reading off a sign
Why: One line of Chapter 6 work.
A subtraction inside the radical always moves the domain to the right, which is worth remembering as a sanity check.
Sorting
Inside or outside the radical.
Sort into buckets
Sort each function by which feature its constant changes.
Three of each, distinguished only by whether the constant sits under the radical sign. That single positional question decides the answer every time.
Prediction
As in the root of x plus two.
Predict first
Where would the domain start?
Correct: At x equal to -2.
\[ x + 2 \ge 0 \;\Longrightarrow\; x \ge -2 \]
Why: The condition is x plus two at least nought, which gives x at least negative two — so the curve starts two units to the left of the origin rather than to the right. Adding inside moves the domain left and subtracting inside moves it right, which feels backwards until you solve the inequality. Testing x equal to negative one confirms it: the radicand is one, whose root is one.
Socratic
It feels like it should move left.
Discussion prompt
Explain why y equals the root of x minus three starts three units to the right. Then say why the intuition points the wrong way.
Hint: What input is needed to get a radicand of nought?
Answer:
The curve starts where the radicand is nought, and x minus three is nought when x is three. So the input has to be three larger than before to produce the same radicand, which pushes every point three units right.
The intuition misleads because the minus sign suggests moving backwards, but the shift is about what x must be rather than what is done to it. Asking what input produces a given radicand — rather than reading the sign — gives the right answer every time, and the same reasoning will apply to every shifted graph in later courses.
Section
Section 5
Concept
A quantity modelled by a square root grows without bound but ever more slowly. Quadrupling the input only doubles the output, which is what distinguishes such a model from a linear one.
Four times the input gives twice the output.
Figure (svg): Walking speed as a square root function of leg length
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-697 — the lesson opener on dinosaur walking speed as a function of leg length
Picture it
Longer legs, but not proportionally faster.
Figure (svg): Walking speed as a square root function of leg length
The curve's flattening is the model's real content. A linear model would predict that a dinosaur with four times the leg length walks four times as fast, and it does not.
Worked example
The lesson opener's situation, with a model supplied.
\[ \text{With } s = 2.5\sqrt{L}, \text{ compare the speeds for leg lengths } 1 \text{ and } 4. \]
Evaluate at one
Why: The root of one is one.
\[ s = 2.5 \]
Evaluate at four
Why: The root of four is two.
\[ s = 5 \]
Compare the inputs
Why: Four times as long.
\[ \times 4 \]
Compare the outputs
Why: Twice as fast.
\[ \times 2 \]
Figure (svg): Walking speed as a square root function of leg length
\[ L: \times 4 \;\Longrightarrow\; s: \times 2 \]
Verify: test the pattern again
Why: At leg length nine the speed is 7.5, which is three times the speed at length one — and nine is nine times one. The output multiplies by the square root of whatever the input multiplies by, which is the defining property of the model.
Faded example
Not by the factor itself.
Fill in the blanks
If the input is multiplied by 9, the output of a square root model is multiplied by 3, because the root of 9 is 3.
Why: The output scales by the square root of the factor applied to the input. That is why such models flatten: large increases in input produce much smaller increases in output.
Worked example
The situation restricts more than the algebra does.
\[ \text{What is the domain of } s = 2.5\sqrt{L} \text{ as a model of walking speed?} \]
Take the algebraic domain
Why: The radicand must be non-negative.
\[ L \ge 0 \]
Consider the situation
Why: A leg has positive length.
\[ L > 0 \]
Consider realism
Why: Legs are not arbitrarily long.
State it sensibly
Why: Within the range of real animals.
Figure (svg): Walking speed as a square root function of leg length
\[ L > 0, \text{ and bounded above in practice} \]
Verify: test an absurd value
Why: At a leg length of ten thousand the model gives a speed of two hundred and fifty, which no animal achieves. The algebra permits it and the situation does not, which is the same distinction drawn about growth models in Chapter 8.
Trap
The leg is four times as long, so the dinosaur walks four times as fast.
Scale the output by the same factor as the input
Why: That is how a direct variation behaves.
The model contains a square root, so the output scales by the square root of the factor. Four times the length gives twice the speed, not four times.
\[ s(4) = 2.5\sqrt{4} = 5 = 2 \cdot s(1) \]
Substitute both values and compare the outputs
Why: Rather than assuming a proportion.
A square root model is not a direct variation, however similar the graphs look near the origin.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Square root model | Direct variation | |
|---|---|---|
| Double the input | output times about 1.41 | output doubles |
| Quadruple the input | output doubles | output quadruples |
| The graph | rises and flattens | a straight line through the origin |
Both pass through the origin and both rise, which is why they are easy to confuse near the start. Their behaviour for large inputs is completely different.
Hypothesis
Speed, period and many other quantities.
Predict first
What do such quantities have in common?
Correct: They grow with size but with diminishing returns.
The same flattening appeared in Lesson 9.3's boat speed model.
Why: A square root model says that making something bigger helps, but each additional unit of size helps less than the last. That describes a great many physical situations — a longer leg does allow a faster walk, and a longer pendulum a slower swing, but neither in direct proportion. Recognising the pattern means recognising when a linear estimate will overstate the benefit of making something larger.
Socratic
It predicts a speed for any leg length.
Discussion prompt
Say what limits the domain of a real square root model beyond the algebra. Then say what a very large input would predict.
Hint: What does the variable actually measure?
Answer:
The situation supplies limits the algebra does not: a leg length must be positive, and it must lie within the range of lengths real animals have. The algebraic domain of all non-negative numbers is far wider than the useful one.
A very large input predicts a very large speed — the model has no ceiling — and at some point that becomes physically impossible for reasons the model knows nothing about, such as the strength of bone. That is the same limitation as the exponential models of Chapter 8, and it is why the range of validity should be stated whenever such a model is used.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Change | Affects | Example |
|---|---|---|
| a constant inside the radical | the domain | root of (x - 3) starts at x = 3 |
| a constant outside the radical | the range | root x + 1 starts at y = 1 |
| a multiplier in front | how steeply it rises | neither the domain nor where it starts |
Three different parts of one expression control three different features. Asking where a number sits relative to the radical sign answers most questions about the graph.
Pattern
To evaluate and graph any function involving a square root, these five moves cover it.
Step one before step two is the order that matters: choosing inputs first risks picking values where the function has no value at all.
OpenStax Intermediate Algebra 2e, §8.7 Use Radicals in Functions §8.7
Check
Only the radicand restricts x.
Check your understanding
What is the domain of y = root x + 1?
Answer: A
Why: The one is outside the radical, so the radicand is still x and the condition is x at least nought.
Check
Solve the radicand's inequality.
Check your understanding
What is the domain of y = the square root of (x - 3)?
Answer: A
Why: The radicand x minus three must be at least nought, which gives x at least three after adding three to both sides.
Check
Read the lowest output.
Check your understanding
What is the range of y = root x + 1?
Answer: A
Why: The smallest value of the root is nought, at x equal to nought, so the smallest output is one and the outputs rise from there.
Real world
This is the dinosaur question from the lesson opener. A dinosaur's maximum walking speed is a function of its leg length, and the function involves a square root.
Discussion prompt
Using the model s equals 2.5 times the square root of L, compare the speeds for leg lengths of 1 and 4 units. Then say what a linear model would have predicted and why the difference matters.
Hint: Substitute both and compare the ratio.
Answer:
\[ s(1) = 2.5\sqrt{1} = 2.5, \qquad s(4) = 2.5\sqrt{4} = 5 \]
Four times the leg length gives twice the speed, because the output scales by the square root of the factor applied to the input.
A linear model would have predicted four times the speed, which is twice too fast — and the error grows with size, since at nine times the leg length the true factor is three rather than nine. That flattening is why very large animals are not proportionally quicker than smaller ones, and it is a genuine biological conclusion drawn from the shape of the function rather than from the data.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is the domain of y equal to the square root of (x - 3)?
Correct: x at least 3.
\[ x - 3 \ge 0 \;\Longrightarrow\; x \ge 3 \]
Why: The domain comes from requiring the radicand to be non-negative, and the radicand here is x minus three rather than x. Solving x minus three at least nought gives x at least three, so the curve begins three units to the right of the origin and nothing exists to the left of that. The first option copies the domain of the basic function without checking what is actually under the radical, which is the commonest slip. The third moves the three across without solving properly — testing x equal to nought settles it, since the radicand would be negative three and has no real root. A subtraction inside the radical always shifts the domain to the right, which feels backwards until the inequality is actually solved.
Explain it
They said the domain of the root of x minus three is x at least nought.
Discussion prompt
In no more than four sentences, explain where the domain actually comes from. Then give them a test that settles it.
Hint: What is under the radical?
Answer:
A usable answer: the restriction comes from the square root refusing negative inputs, so it applies to whatever is written under the radical sign. Here that is x minus three, so the condition is x minus three at least nought, which gives x at least three.
The test is to try a value between nought and three, such as one. The radicand becomes negative two, which has no real square root, so the function has no value there and the domain cannot start at nought.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The domain is fixed by writing the radicand's inequality and solving it rather than copying a previous answer. Inside or outside is fixed by looking at what the radical sign covers. The range is fixed by finding the smallest or largest output, usually at the domain's endpoint. Table values are fixed by choosing inputs that make the radicand a perfect square. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page sketch the basic square root function, shade its domain along the horizontal axis and its range up the vertical one, and write beside each why it starts where it does. Underneath, take four functions — one with a multiplier, one with a constant added outside, one with a constant subtracted inside, and one with the root subtracted — and for each write the radicand's inequality, the domain, and the range, in three columns. In the middle, graph two of them on the same axes so the difference between a vertical shift and a horizontal one is visible, marking the starting point of each curve. Beneath that, build a table for a function with a shifted domain by choosing inputs that make the radicand a perfect square, and note the spacing between your chosen inputs. In the lower corner, evaluate a square root model at one, four and nine, write the ratio of outputs beside the ratio of inputs, and say in one sentence why a linear estimate would overstate the effect of increasing the input. Finally, in the margin, write the one question that decides whether a constant moves the domain or the range.
Your chosen table inputs for a shifted domain should be spaced by the odd numbers. If they are evenly spaced, you picked convenient x values rather than convenient radicands, and the outputs will need rounding.
Recap
Five things, and the first governs everything else.
| If the question says | Your first move is |
|---|---|
| Find the domain | Set the radicand at least zero and solve |
| A constant is inside the radical | It moves the domain |
| A constant is outside the radical | It moves the range |
| Make a table | Choose inputs making the radicand a square |
| A model contains a square root | Expect flattening, not proportionality |
Lesson 12.2 works with radical expressions themselves rather than with functions built from them. Adding, subtracting and multiplying radicals brings back the simplifying of Lesson 9.3, with the distributive property in play.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.1 Functions Involving Square Roots §12.1, pp. 692-697 — everything on these slides traces back here
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