The two special cases of the standard form: when A is zero the equation reduces to y equals a constant and graphs as a horizontal line, and when B is zero it reduces to x equals a constant and graphs as a vertical line. Includes writing the equation of such a line from its graph, constant functions with their domain and range, and why a vertical line is not a function.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 4 — Graphing Linear Equations and Functions
Graphing Horizontal and Vertical Lines
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. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-221 — the lesson these objectives are drawn from
Warm-up
Lesson 4.2 built tables from equations with two variables. This lesson asks what happens when one of them goes missing.
Discussion prompt
In the equation y equals 2 there is no x at all. Choose three different values of x and work out y for each. What do you notice, and what does that mean for the graph?
Hint: Try substituting your values and see where the x would go.
Answer:
\[ x = -3 \rightarrow y = 2, \quad x = 0 \rightarrow y = 2, \quad x = 3 \rightarrow y = 2 \]
There is nowhere to put the x, so it never affects the answer and y stays at two whatever you choose. That gives the points negative three comma two, zero comma two and three comma two — all at the same height, so the graph is a horizontal line.
Concept
Every linear equation can be written as Ax plus By equals C. When A is zero the equation reduces to y equals a constant and its graph is a horizontal line; when B is zero it reduces to x equals a constant and its graph is a vertical line.
horizontal line — The graph of an equation of the form y equals a constant, on which every point shares the same y-coordinate.
The variable that has disappeared is the one left free; the one that remains is pinned to a single value.
Figure (svg): The standard form with A zero and with B zero, giving the two special cases
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216
Section
Section 1
Concept
The standard form allows either coefficient to be zero, provided they are not both zero. Each case removes one variable from the equation and produces one of two special kinds of line.
\[ y = 2 \;\Longleftrightarrow\; 0x + 1y = 2 \]
Figure (svg): The standard form with A zero and with B zero, giving the two special cases
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216 — the paragraph on A equals zero and B equals zero
Picture it
One coefficient set to zero, twice.
Figure (svg): The standard form with A zero and with B zero, giving the two special cases
Both remain linear equations, so both graph as straight lines. The special thing about them is which direction those lines run, not that they are exceptions to anything.
Worked example
Seeing the hidden zero coefficient makes these equations ordinary.
\[ \text{Write } \; y = 2 \; \text{ and } \; x = -3 \; \text{ in the form } Ax + By = C. \]
Write y equals 2 with both variables shown
Why: The x term has a coefficient of zero, which is why it is not written.
\[ 0 x + 1 y = 2 \]
Read off the coefficients
Why: A is zero, B is one, C is two.
\[ A = 0, B = 1 \]
Write x equals negative 3 the same way
Why: This time the y term has the zero coefficient.
\[ 1 x + 0 y = -3 \]
Check the condition
Why: In each case only one coefficient is zero, so the standard form is satisfied.
Figure (svg): The solution to Worked example rewrite in standard form shown as a ladder of expressions, one row per algebraic move
\[ 0x + 1y = 2 \qquad 1x + 0y = -3 \]
Verify: substitute a pair into the expanded form
Why: For y equals 2 at the point (5, 2): zero times five plus one times two is two, which matches. The zero coefficient makes the x-value irrelevant, which is exactly what the graph shows by running across every value of x.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216
Sorting
Look at which variables appear.
Sort into buckets
Sort each equation by the kind of line it graphs.
The test is simply which variables are present. Nothing about the numbers matters, and no table needs building to answer it.
Worked example
The variable that disappears is the one left unconstrained.
\[ \text{In } y = 2 \text{ and } x = -3, \text{ say which variable is fixed and which is free.} \]
Look at y equals 2
Why: The equation says something about y and nothing about x.
Say what that means for x
Why: Any value of x satisfies the equation, so x is free.
Look at x equals negative 3
Why: The equation constrains x and says nothing about y.
Say what that means for y
Why: Any value of y works, so y is free.
Figure (svg): The solution to Worked example which variable is free shown as a ladder of expressions, one row per algebraic move
\[ y = 2: \; y \text{ fixed}, \; x \text{ free} \qquad x = -3: \; x \text{ fixed}, \; y \text{ free} \]
Verify: check against the direction of each line
Why: The free variable is the one that ranges over every value, and the line runs in that variable's direction. With x free the line runs horizontally; with y free it runs vertically. The picture and the algebra agree.
Trap
\[ y = 2 \]
Conclude that x must be zero, since no x appears
Why: An absent letter looks like an absent quantity, so zero seems like the natural value.
That would give the single point zero comma two rather than a whole line. The equation places no restriction on x at all, which is the opposite of forcing it to zero.
\[ y = 2 \;\Longleftrightarrow\; 0x + 1y = 2 \]
Read a missing variable as unconstrained rather than as zero
Why: A coefficient of zero means the term contributes nothing, so the variable may take any value whatever.
Writing the hidden zero coefficient makes this visible: zero times anything is zero, so no choice of x can affect the equation.
Faded example
Write each equation with both variables present.
Fill in the blanks
y = 2 \;\Longleftrightarrow\; 0x + 1y = 2 \qquad x = -3 \;\Longleftrightarrow\; 1x + 0y = -3
Why: Each equation hides a coefficient of zero on the variable it does not mention. Writing that zero explicitly shows why the variable is free: zero times any value is zero, so the equation is unaffected by what that variable does.
Elimination
Standard form is Ax plus By equals C with A and B not both zero.
Eliminate the wrong options
Which of these fails the standard form?
Survives elimination: A
Why: With both coefficients zero the left side is always zero, so the equation reads zero equals five — a false statement mentioning neither variable. The standard form excludes it precisely because it describes no line at all, which is why the condition not both zero appears in the definition.
Socratic
It would be simpler to require both coefficients to be non-zero.
Discussion prompt
Explain what would be lost if the standard form insisted that neither A nor B could be zero. Then say which two very familiar lines would be excluded.
Hint: Think about the axes themselves.
Answer:
Horizontal and vertical lines would stop counting as linear, even though they are straight and satisfy every other property of lines. A definition that excludes some straight lines from being called linear would be an awkward one, and every later theorem would need an exception clause.
The two most familiar casualties would be the axes themselves: the x-axis is the line y equals zero and the y-axis is x equals zero. Excluding them would mean the coordinate plane's own axes were not linear graphs, which is clearly the wrong definition.
Section
Section 2
Concept
The equation y equals b has no x in it, so the y-coordinate is always b regardless of x. The graph is a horizontal line b units above the x-axis, or below it if b is negative.
\[ y = 2 \;\Longrightarrow\; (-3, 2), \; (0, 2), \; (3, 2), \; \ldots \]
Every ordered pair whose second coordinate is b is a solution, and nothing else is.
Figure (svg): The horizontal line y equals 2 with three of its points marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216 — Example 1, Graph the Equation y = b
Picture it
Three of its infinitely many points.
Figure (svg): The horizontal line y equals 2 with three of its points marked
The three marked points have completely different x-coordinates and identical y-coordinates. That shared height is what the equation is asserting, and it is what makes the line horizontal.
Worked example
This is Example 1 from the textbook.
\[ \text{Graph the equation } \; y = 2. \]
Notice that x does not appear
Why: The equation says nothing about x, so x may take any value.
Generate some solutions
Why: Choose any x-values and pair each with y equal to two.
\[ (-3, 2), (0, 2), (3, 2) \]
Plot the points
Why: All three sit at the same height, two units above the x-axis.
Draw the line through them
Why: A horizontal line, extended in both directions.
Figure (svg): The horizontal line y equals 2 with three of its points marked
\[ y = 2 \text{ is horizontal, } 2 \text{ units above the } x\text{-axis} \]
Verify: test a point far from those plotted
Why: The pair (100, 2) should satisfy the equation, and it does — the y-coordinate is two. The line really does extend indefinitely in both directions, and testing a distant point confirms that the drawing's arrows are honest.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216
Sorting
Only the second coordinate matters.
Sort into buckets
Sort each pair by whether it lies on the line y equals 2.
Two of the rejected pairs contain a two in the first position, which is exactly the trap. The equation constrains y, so only the second coordinate is consulted.
Worked example
Guided Practice 1 to 3. One is negative and one is fractional.
\[ \text{Graph } \; y = 3, \quad y = -4, \quad y = \tfrac{1}{2}. \]
Graph y equals 3
Why: A horizontal line three units above the x-axis.
\[ 3\text{ above} \]
Graph y equals negative 4
Why: A horizontal line four units below the x-axis, since the constant is negative.
\[ 4\text{ below} \]
Graph y equals one half
Why: A horizontal line half a unit above the x-axis.
\[ 0.5\text{ above} \]
Note what changed between them
Why: Only the height. All three are horizontal, and the constant places each one.
Figure (svg): The solution to Worked example three from guided practice shown as a ladder of expressions, one row per algebraic move
\[ y = 3, \; y = -4, \; y = \tfrac{1}{2} \]
Verify: check the sign of each height against the axis
Why: The positive constants put lines above the x-axis and the negative one puts a line below it, exactly as a positive or negative y-coordinate did for a single point in Lesson 4.1. Nothing new about signs is happening here.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-216
Error analysis
The student graphed three equations. Two are wrong.
Annotate
On: \( \begin{aligned} y = 2 &: \; \text{vertical line 2 units right} \\ y = -4 &: \; \text{horizontal line 4 units below} \\ y = 3 &: \; \text{the single point } (0, 3) \end{aligned} \)
The two errors are the two standard ones: swapping the direction, and mistaking an unconstrained variable for a fixed one. Generating three solutions before drawing catches both.
Faded example
Pair any three x-values with the fixed y.
Fill in the blanks
y = -4: \quad (-2, -4), \; (0, -4), \; (7, -4)
Why: All three blanks hold the same number, which is the entire content of the equation: y is negative four no matter what x does. Writing three identical outputs beside three different inputs is what makes the horizontal direction obvious.
Prediction
The constant places the line and the sign chooses the side.
Predict first
Where is the graph of y equals negative 4?
Correct: A horizontal line 4 units below the x-axis.
\[ y = -4: \; (-2, -4), \; (0, -4), \; (7, -4), \; \ldots \]
Why: The equation fixes y at negative four, so every point sits four units below the horizontal axis and the line runs across. A negative constant puts the line below, exactly as a negative y-coordinate put a point below in Lesson 4.1, and the absence of x means infinitely many points rather than one.
Socratic
The equation mentions y, and the line runs in the x direction.
Discussion prompt
Explain why fixing the y-coordinate produces a line running horizontally, which feels backwards to many people. Then say what the graph would be if both coordinates were fixed.
Hint: Ask which coordinate is allowed to vary.
Answer:
Fixing y stops any vertical movement, so the points cannot spread up or down. What they can do is spread left and right, because x is unconstrained — and a set of points spread horizontally is a horizontal line. The equation names the coordinate that is stuck, and the line runs in the direction of the one that is free.
If both were fixed, as in the pair of conditions x equals 2 and y equals 3, nothing could vary at all and the graph would be the single point (2, 3). That is what a solution of a system looks like, and Chapter 7 is entirely about finding such points.
Section
Section 3
Concept
The equation x equals a has no y in it, so the x-coordinate is always a regardless of y. The graph is a vertical line a units to the right of the y-axis, or to the left if a is negative.
\[ x = -3 \;\Longrightarrow\; (-3, -2), \; (-3, 0), \; (-3, 3), \; \ldots \]
This is the mirror image of the previous section, with the roles of the two variables exchanged.
Figure (svg): The vertical line x equals negative 3 with three of its points marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217 — Example 2, Graph the Equation x = a
Picture it
Three points sharing an x-coordinate.
Figure (svg): The vertical line x equals negative 3 with three of its points marked
The three points have completely different heights and identical horizontal positions. Everything about this section is the previous one with x and y swapped.
Worked example
This is Example 2 from the textbook.
\[ \text{Graph the equation } \; x = -3. \]
Notice that y does not appear
Why: The equation says nothing about y, so y may take any value.
Generate some solutions
Why: Pair x equal to negative three with any y-values.
\[ (-3, -2), (-3, 0), (-3, 3) \]
Plot the points
Why: All three sit three units to the left of the y-axis.
Draw the line through them
Why: A vertical line, extended up and down.
Figure (svg): The vertical line x equals negative 3 with three of its points marked
\[ x = -3 \text{ is vertical, } 3 \text{ units left of the } y\text{-axis} \]
Verify: test a point far from those plotted
Why: The pair (-3, 50) should satisfy the equation, and it does, since its x-coordinate is negative three. The line extends indefinitely upwards and downwards, and a distant point confirms it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217
Discrimination
The variable that is missing is the one that varies.
Sort into buckets
Sort each equation by the direction of its line.
Worked example
Guided Practice 4 to 6. One is negative and one is fractional.
\[ \text{Graph } \; x = 2, \quad x = -1, \quad x = 3\tfrac{1}{2}. \]
Graph x equals 2
Why: A vertical line two units right of the y-axis.
\[ 2\text{ right} \]
Graph x equals negative 1
Why: A vertical line one unit left of the y-axis.
\[ 1\text{ left} \]
Graph x equals three and a half
Why: A vertical line three and a half units right, between the marks at 3 and 4.
\[ 3.5\text{ right} \]
Note what changed between them
Why: Only the horizontal position. All three are vertical.
Figure (svg): The solution to Worked example three from guided practice shown as a ladder of expressions, one row per algebraic move
\[ x = 2, \; x = -1, \; x = 3\tfrac{1}{2} \]
Verify: check each against the y-axis
Why: Positive constants put lines to the right of the y-axis and the negative one to the left, exactly as a positive or negative x-coordinate placed a point in Lesson 4.1. And a fractional constant sits between two grid lines, which is perfectly ordinary.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217
Trap
\[ x = -3 \]
Draw a horizontal line, since the equation mentions x and x is the horizontal axis
Why: The letter x is associated with the horizontal direction, so the line seems to belong there.
The equation fixes the horizontal position rather than describing horizontal movement. A horizontal line through the left of the plane would have equation y equals something instead.
\[ x = -3 \text{ is a vertical line} \]
Generate two solutions before drawing anything
Why: The pairs (-3, 0) and (-3, 3) have different heights and the same horizontal position, so the line joining them must be vertical.
Plotting two points is faster than reasoning about which direction the letter suggests, and it cannot be misremembered.
Faded example
Pair the fixed x with any three y-values.
Fill in the blanks
x = 2: \quad (2, -1), \; (2, 0), \; (2, 4)
Why: Every solution has an x-coordinate of two and any y-coordinate at all. The three identical first coordinates with three different second ones is what forces the line to run vertically, and it is worth writing out once rather than remembering a rule.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| y = b | x = a | |
|---|---|---|
| Which variable is fixed | y | x |
| Which variable is free | x | y |
| Direction of the line | horizontal | vertical |
| The axis it is parallel to | the x-axis | the y-axis |
Every row swaps between the two columns. The two cases are mirror images, so learning one and knowing it reverses is enough.
Socratic
Lesson 4.2 rewrote every equation as y equals something. This one cannot be.
Discussion prompt
Explain why the equation x equals negative 3 cannot be written in function form. Then say what that suggests about whether it defines y as a function of x.
Hint: Function form isolates y on one side.
Answer:
Function form means y isolated on one side, and there is no y anywhere in x equals negative three to isolate. No rearrangement can produce one, since the equation genuinely says nothing about y.
That suggests the equation does not determine y from x, and it does not: at x equal to negative three, y can be anything at all. The next section makes this precise — a vertical line fails the definition of a function from Lesson 1.8, and the impossibility of writing it in function form is the algebraic symptom of that failure.
Section
Section 4
Concept
To write the equation of a horizontal or vertical line, look at several of its points and find the coordinate they all share. Set that variable equal to the shared value.
A horizontal line shares its y-coordinate, and a vertical one shares its x-coordinate.
Figure (svg): Two graphed lines with their equations read off from a shared coordinate
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217 — Example 3, Write an Equation of a Line
Picture it
Read the shared coordinate off each.
Figure (svg): Two graphed lines with their equations read off from a shared coordinate
The left line's points all have x equal to negative two, and the right line's all have y equal to negative four. Reading two points and comparing is the whole method.
Worked example
This is Example 3 from the textbook. One vertical line and one horizontal.
\[ \text{A vertical line passes through } (-2, 1) \text{ and } (-2, -1). \text{ A horizontal one passes through } (-1, -4) \text{ and } (2, -4). \]
Compare the coordinates of the first pair
Why: The x-coordinates are both negative two; the y-coordinates differ.
Write the first equation
Why: Set x equal to the shared value.
\[ x = -2 \]
Compare the coordinates of the second pair
Why: The y-coordinates are both negative four; the x-coordinates differ.
Write the second equation
Why: Set y equal to the shared value.
\[ y = -4 \]
Figure (svg): Two graphed lines with their equations read off from a shared coordinate
\[ x = -2 \qquad y = -4 \]
Verify: check a third point on each line
Why: Any point three units above the first pair, such as (-2, 4), still has x equal to negative two, so it satisfies the equation. The same holds for the horizontal line at any x. A third point confirms the shared coordinate is genuinely shared rather than a coincidence of two.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217
Matching
Find the coordinate the two points share.
Match the pairs
Why: In each pair one coordinate is repeated and the other differs, and the equation names the repeated one. Two of these give vertical lines and two horizontal, and the only thing distinguishing them is which position the repetition sits in.
Worked example
Guided Practice 7 and 8. Read the shared coordinate in each.
\[ \text{One line passes through } (1, 1) \text{ and } (1, -3). \text{ Another passes through } (-2, 3) \text{ and } (3, 3). \]
Compare the first pair
Why: Both x-coordinates are one, and the y-coordinates differ.
\[ x = 1 \]
Name the direction
Why: A shared x-coordinate means a vertical line.
Compare the second pair
Why: Both y-coordinates are three, and the x-coordinates differ.
\[ y = 3 \]
Name the direction
Why: A shared y-coordinate means a horizontal line.
Figure (svg): The solution to Worked example two from guided practice shown as a ladder of expressions, one row per algebraic move
\[ x = 1 \text{ (vertical)} \qquad y = 3 \text{ (horizontal)} \]
Verify: check that the non-shared coordinates really do differ
Why: In the first pair the y-values are one and negative three, which differ, confirming that y is free. Had both coordinates matched, the two points would be the same point and no line would be determined.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 217-217
Trap
A line passes through (-2, 1) and (-2, -1).
Notice that the y-values are 1 and -1 and write y = 1 or y = -1
Why: The differing coordinates are the ones that catch the eye, since they are the ones with something to compare.
Neither equation describes the line: y equals 1 is a horizontal line through only one of the two points, and y equals negative 1 through only the other.
\[ x = -2 \]
Name the coordinate that is the same, not the one that changes
Why: The equation states what is true of every point on the line, and only the shared coordinate is true of all of them.
Testing the proposed equation on both given points settles it immediately: x equals negative two holds for both, and y equals one holds for only one.
Faded example
Identify the shared coordinate and set it equal to its value.
Fill in the blanks
\textx (5, -2) \text5 (5, 4) \;\Longrightarrow\; ___ = ___
Why: The x-coordinates are both five while the y-coordinates differ, so x is the shared coordinate and the equation is x equals five. The line is vertical, passing five units to the right of the y-axis.
Elimination
A line passes through (3, -1) and (3, 5).
Eliminate the wrong options
Which equation describes it?
Survives elimination: A
Why: Both points have an x-coordinate of three, so x equals three is true of both and of every other point on the vertical line through them. Testing a candidate equation against both given points is the reliable check, and it rejects the other three at once.
Socratic
Lesson 4.2 recommended a third point as a check.
Discussion prompt
Explain why two points are enough to identify a horizontal or vertical line's equation, and say what could still go wrong that a third point would catch.
Hint: Think about how much freedom the equation has.
Answer:
These equations have only one number in them, and a single point already determines it — the second point serves mainly to confirm which coordinate is the shared one. Two points with a repeated coordinate settle both the direction and the value at once.
What a third point could still catch is a misread coordinate. If one of the two points was read wrongly off the graph, the repetition might be an accident, and a third point on the line would disagree. That is the same argument as in Lesson 4.2, and it costs a few seconds.
Section
Section 5
Concept
A horizontal line pairs every input with exactly one output, so it is a function — a constant function, since the output never changes. A vertical line pairs one input with infinitely many outputs, so it is not a function at all.
constant function — A function whose output is the same for every input. Its graph is a horizontal line, and its range contains a single value.
Figure (svg): An input-output table for y equals 2, showing every input giving the same output
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-218 — the key words constant function, domain and range
Picture it
Six inputs, all giving the same output.
Figure (svg): An input-output table for y equals 2, showing every input giving the same output
Two inputs sharing an output is permitted by the definition from Lesson 1.8, and here every input shares it. The domain is everything and the range is a single number.
Worked example
The definitions from Lesson 1.8, applied to the simplest possible function.
\[ \text{Give the domain and range of } \; y = 2. \]
Ask which inputs are allowed
Why: Any value of x gives a valid solution, since x is unconstrained.
Ask which outputs occur
Why: Every input produces the output two, and nothing else ever occurs.
Check the function requirement
Why: Each input has exactly one output, which the definition requires.
Name it
Why: A function whose output never changes is called a constant function.
Figure (svg): An input-output table for y equals 2, showing every input giving the same output
\[ \text{domain: all reals} \qquad \text{range: } \{2\} \]
Verify: check against the definition's asymmetry
Why: Many inputs share the output two, which Lesson 1.8 explicitly permits — the definition restricts arrows leaving an input, never arrows arriving at an output. A constant function is the extreme case of that permission, and it is a function precisely because of it.
Sorting
Check whether any input has more than one output.
Sort into buckets
Sort each equation by whether it defines y as a function of x.
Every vertical line fails and every other line passes. That is the whole content of the vertical line test, which Chapter 4 will state formally.
Worked example
Applying the same definition to x equals negative three.
\[ \text{Explain why } \; x = -3 \; \text{ is not a function of } x. \]
Ask what output the input negative 3 has
Why: The pairs (-3, 0), (-3, 1) and (-3, 5) are all solutions.
Apply the definition
Why: A function requires each input to have exactly one output, and this input has infinitely many.
Ask about other inputs
Why: No other input has any output at all, since no other x-value satisfies the equation.
State the conclusion
Why: The pairing is not a function of x.
Figure (svg): The vertical line x equals negative 3 shown failing the one-output requirement
\[ x = -3: \; (-3, 0), (-3, 1), (-3, 5), \ldots \]
Verify: ask the rule its own question
Why: What is the output when the input is negative three? There is no single answer, and a rule that cannot answer its own question is exactly what the definition in Lesson 1.8 excludes. This is the same failure as the pairing 2 to 5 and 2 to 7 in that lesson.
Trap
\[ y = 2 \]
Rule it out because every input gives the same output
Why: A function that never varies looks degenerate, and the word function suggests something that does something.
Sharing an output among inputs is explicitly permitted. What the definition forbids is one input having two outputs, which never happens here.
y = 2 is a function — a constant function.
Check inputs, never outputs, exactly as in Lesson 1.8
Why: The requirement is that each input has exactly one output, and here each has exactly one, namely two.
The vertical line is the one that fails, and it fails in the other direction: one input with many outputs rather than many inputs with one.
Elimination
The line x equals negative 3 fails the definition from Lesson 1.8.
Eliminate the wrong options
What is the reason?
Survives elimination: A
Why: The definition requires each input to have exactly one output, and negative three has endlessly many. Option B describes the same picture from the wrong side, and getting that direction right is the single most common difficulty with the definition, exactly as it was in Lesson 1.8.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| y = 2 | x = -3 | |
|---|---|---|
| A function of x? | Yes | No |
| Domain | all real numbers | the single value -3 |
| Range | the single value 2 | all real numbers |
The domain and range swap between the two columns, which is what you would expect from two mirror-image cases. Only one of them satisfies the definition, and the asymmetry of that definition is why.
Socratic
The definition is about inputs and outputs, and a graph shows points.
Discussion prompt
Invent a test you could apply to any graph to decide whether it defines y as a function of x, using only the picture. Then apply it to a horizontal line and to a vertical one.
Hint: Think about what one input looks like on the picture.
Answer:
One input is one x-value, which on the picture is a vertical line drawn at that position. The graph gives that input more than one output exactly when the vertical line meets it more than once. So the test is: if any vertical line crosses the graph more than once, it is not a function.
A horizontal line is crossed exactly once by every vertical line, so it passes. A vertical line is met infinitely often by the vertical line in the same position, so it fails. This is the vertical line test, and you have just derived it from the definition rather than being handed it — which is worth doing, since it is easy to misremember which way round it goes.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Equation | Which variables appear | The graph |
|---|---|---|
| y = b | y only | a horizontal line |
| x = a | x only | a vertical line |
| y = mx + b | both | a slanted line |
Which variables appear settles the direction before any table is built or any point is plotted.
Pattern
Whether you are graphing such a line or reading its equation off a picture, the same five moves cover it.
Step three is worth doing even when the answer feels obvious. Two plotted points settle the direction without relying on remembering which letter goes with which axis.
OpenStax Elementary Algebra 2e, §4.2 Graph Linear Equations in Two Variables §4.2
Check
Read which variable appears.
Check your understanding
What is the graph of y equals negative 5?
Answer: A
Why: The equation fixes y at negative five and says nothing about x, so every point sits five units below the horizontal axis and the line runs across. Solutions include (-2, -5), (0, -5) and (7, -5).
Check
Find the shared coordinate.
Check your understanding
A line passes through (4, -3) and (4, 2). What is its equation?
Answer: A
Why: Both points have an x-coordinate of four while their y-coordinates differ, so x is the shared coordinate and the line is vertical through x equals four. Testing both points against the equation confirms it.
Check
Apply the definition from Lesson 1.8.
Check your understanding
Which of these is NOT a function of x?
Answer: A
Why: The vertical line x equals 2 pairs the single input two with infinitely many outputs, so no single output can be named and the definition fails. Every other option gives each input exactly one output.
Real world
A thermostat holds a room at 20 degrees all day. Separately, a shop closes permanently at 6pm on one particular date, so nothing is sold after that instant.
Discussion prompt
Sketch each situation as a graph with time across, and say which one is a horizontal line and which a vertical one. Then say which of the two is a function of time and what that means practically.
Hint: Ask what is being held constant in each case.
Answer:
The thermostat gives a horizontal line at height twenty: for every time, the temperature is twenty degrees. It is a constant function of time, with the whole day as its domain and the single value twenty as its range.
The closing instant gives a vertical line at that one time: the event occupies a single moment and no others. As a graph of something against time it is not a function of time, since that one input would have to pair with every value on the vertical axis at once.
Practically, the difference is that you can ask the thermostat's graph what the temperature is at any moment and get one answer, while the vertical line answers no such question — it marks a moment rather than describing a quantity over time. Vertical lines in real graphs almost always mark boundaries rather than measurements.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is the equation y equals 2 a linear equation?
Correct: Yes, it is 0x plus 1y equals 2 in standard form.
\[ y = 2 \;\Longleftrightarrow\; 0x + 1y = 2 \quad A = 0, \; B = 1, \; C = 2 \]
\[ x = -3 \;\Longleftrightarrow\; 1x + 0y = -3 \quad \text{also linear} \]
Why: The standard form requires only that A and B are not both zero, and here A is zero while B is one. The equation is linear, its graph is a straight line, and the fact that the line happens to be horizontal is not a disqualification. Excluding these would also exclude the x-axis and the y-axis from being linear graphs, which is clearly the wrong definition.
Explain it
They can graph y equals 3x minus 2 and are baffled by y equals 2.
Discussion prompt
In no more than four sentences, explain what the graph of y equals 2 looks like and why, without stating a rule for them to memorise. Then give them the trick for telling which of the two special cases is which.
Hint: Have them generate solutions rather than recall a rule.
Answer:
A usable answer: pick any three values of x and work out y for each. There is nowhere to put the x, so y comes out as two every time, giving points at the same height but different positions across. Points at the same height make a horizontal line.
The trick is to plot two solutions rather than trying to remember which letter means which direction. Two points with the same y and different x can only be joined horizontally, and two with the same x and different y can only be joined vertically — the picture tells you, so nothing has to be memorised.
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 direction is fixed by plotting two solutions rather than recalling a rule. Unconstrained-versus-zero is fixed by writing the hidden zero coefficient explicitly. Reading an equation off a graph is fixed by naming the coordinate that repeats rather than the one that changes. The function question is fixed by checking inputs and never outputs, exactly as in Lesson 1.8. 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
Draw one coordinate plane and on it graph three horizontal lines and three vertical lines, labelling each with its equation and including at least one negative and one fractional constant in each family. Beside the plane, write each equation again in the expanded form with its hidden zero coefficient shown. Underneath, write two points from one of your vertical lines and two from one of your horizontal lines, circling in each pair the coordinate that repeats and writing the equation it gives. In the lower half, make an input-output table for one of your horizontal lines with six different inputs, and write its domain and range beside it. Finally, in the margin, write one sentence explaining why one family is made of functions and the other is not.
Your horizontal-line table should show six different inputs and six identical outputs. If any two outputs differ, check whether you have accidentally tabulated a slanted line instead.
Recap
Five things, and the second is the one that feels backwards until you plot two points.
| If the question says | Your first move is |
|---|---|
| Graph y = 2 | Pair three different x-values with y equal to 2 |
| Graph x = -3 | Pair three different y-values with x equal to -3 |
| Write the equation of the line | Find the coordinate that repeats |
| Is it a function | Ask whether any input has two outputs |
| Give the domain and range | Ask which values each variable is allowed |
Lesson 4.4 returns to slanted lines and finds the two points that are easiest of all to compute: the places where the line crosses each axis, which turn out to make graphing much quicker than a table.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 4 Graphing Linear Equations and Functions — Lesson 4.3 Graphing Horizontal and Vertical Lines §4.3, pp. 216-221 — everything on these slides traces back here
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