Defines the parabola by equal distances to a focus and a directrix, connects that definition to the quadratic graphs of chapter 3, identifies the vertex, focus, directrix and axis from a standard equation in all four orientations, and derives the reflection property.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 10 — Analytic Geometry
§10.3 The Parabola, pp. 1230-1247
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1230-1247 — the pages these objectives are drawn from
Warm-up
Chapter 3 graphed quadratics without ever mentioning a focus.
Discussion prompt
The graph of a squared function is a parabola. What was never said about it there?
Hint: What defines the curve, rather than describing it?
Answer:
Chapter 3 gave a formula and drew its graph. Nothing was said about why that shape rather than another.
The other two conics were defined by a distance condition on two foci, and their equations followed. The parabola has such a definition too.
It uses one focus and one line, called the directrix, and requires the distances to them to be equal. The familiar quadratic graph is exactly the curve that condition produces.
Concept
A parabola is the set of points equidistant from a fixed point, the focus, and a fixed line, the directrix. The vertex sits halfway between them.
directrix — the fixed line from which a parabola's points are as far as they are from the focus
\[ y^2=4px \quad\text{or}\quad x^2=4py \]
The single constant p is the distance from the vertex to the focus, and also from the vertex to the directrix. Everything about the parabola is determined by it, where the other conics needed two constants.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1230-1234
Section
Section 1
Concept
Every point of a parabola is as far from the focus as it is from the directrix. That single condition determines the whole curve.
Distance to a line means perpendicular distance, which is why the dropped segment in the diagram is drawn at a right angle. Measuring to the nearest point of the line is what makes the condition well defined.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1230-1235
Picture it
Two equal distances, one to a point and one to a line.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
At the vertex the two distances are both the focal distance, which is why the vertex sits exactly halfway between the focus and the directrix.
Worked example
Halfway between focus and directrix.
\[ \text{A parabola has focus } (0,3) \text{ and directrix } y=-3. \text{ Where is the vertex?} \]
Note the axis
Why: Perpendicular to the directrix through the focus.
Find the midpoint
Why: Between the focus and the line.
Compute
Why: Average the two heights.
\[ y = 0 \]
State the vertex
Why: On the axis.
\[ (0, 0) \]
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
\[ (0,0) \]
Verify: check the two distances
Why: From the origin to the focus is 3, and to the directrix is also 3 — equal, as the definition requires. The vertex is always the point where those two distances are smallest, and they are equal there as everywhere else on the curve.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1231-1233
Prediction
The focus is 4 units from the directrix.
Predict first
How far is the vertex from each?
Correct: Two units from each.
Why: The vertex is on the curve, so it is equidistant from focus and directrix — and it lies between them on the axis, so each distance is half the gap. That half-distance is the constant p.
Worked example
Compute both distances.
\[ \text{Is } (6,3) \text{ on the parabola with focus } (0,3) \text{ and directrix } y=-3? \]
Find the distance to the focus
Why: Horizontal separation only.
\[ 6 \]
Find the distance to the directrix
Why: Vertical, to the line.
\[ 3 + 3 = 6 \]
Compare
Why: Both are six.
Conclude
Why: The condition holds.
Figure (svg): The solution to Worked example verify a point is on the curve shown as a ladder of expressions, one row per legal move
\[ \text{on the parabola} \]
Verify: check against the equation
Why: With p equal to 3 and a vertical axis, the equation is x squared equals twelve y — and 36 equals twelve times 3, which holds. The definition and the equation agree, as they must.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1233-1235
Trap
\[ \text{measure from the point to some convenient spot on the directrix} \]
Take any segment reaching the line
Why: Distance to a line is treated like distance to a point.
The measurement is too long, and points on the curve appear to fail the condition.
Distance to a line means perpendicular distance, measured to the nearest point of the line.
Any other segment is longer, so using one would break the equidistance condition for points that genuinely satisfy it.
For a horizontal directrix that means a purely vertical drop, which makes the computation a single subtraction.
Sorting
Each definition uses different data.
Sort into buckets
Sort each defining condition.
Faded example
For a point 6 units from the focus.
Fill in the blanks
\text6=3+3=6, \text______
Why: The perpendicular distance to a horizontal directrix is the vertical separation, computed as a single subtraction. Equality with the focal distance is what places the point on the curve.
Explain it to yourself
The other two conics have centres and the parabola does not.
Discussion prompt
Explain what accounts for the difference.
Hint: What would a centre be midway between?
Answer:
An ellipse's or hyperbola's centre is the midpoint of the two foci. That construction needs two points of the same kind.
A parabola has one focus and one line, which are different kinds of object with no midpoint between them in the same sense.
What it has instead is a vertex — the point on the curve nearest both. A good explanation notes that fewer defining objects means fewer features, which is also why there is no second axis and no eccentricity ratio to compute.
Section
Section 2
Concept
Whichever variable is squared is the one that does not determine the direction; the curve opens along the other axis, in the direction fixed by the sign of the constant.
Two independent choices — which variable is squared and the sign of the constant — give exactly four orientations. That is why the table has four rows and not more.
| squared variable | sign of p | opens |
|---|---|---|
| the vertical one | positive | rightward |
| the vertical one | negative | leftward |
| the horizontal one | positive | upward |
| the horizontal one | negative | downward |
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1235-1239
Picture it
Two choices, four outcomes.
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
The unsquared variable is the one that can grow without bound, which is the axis the curve opens along. The squared variable is symmetric about the axis instead.
Worked example
Find the squared variable and the sign.
\[ \text{Describe } y^2=-8x. \]
Find the squared variable
Why: The vertical one.
Read the constant
Why: Four p equals negative eight.
\[ p = -2 \]
Read the direction
Why: Negative means leftward.
Locate the focus
Why: Two units left of the vertex.
\[ (-2, 0) \]
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
\[ \text{opens left},\; \text{focus }(-2,0),\; \text{directrix }x=2 \]
Verify: check a point
Why: At x equal to negative 2 the equation gives y squared equal to 16, so y is plus or minus 4. Those two points are each 4 units from the focus vertically and 4 units from the directrix horizontally — equidistant, as required.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1236-1238
Prediction
The horizontal variable is squared.
Predict first
Which way does the parabola open?
Correct: Vertically.
Why: The squared variable is symmetric about the axis rather than along it, so the curve opens along the other one. The sign then decides whether that is upward or downward.
Worked example
The one chapter 3 always drew.
\[ \text{Describe } x^2=12y. \]
Find the squared variable
Why: The horizontal one.
Read the constant
Why: Four p equals twelve.
\[ p = 3 \]
Read the direction
Why: Positive means upward.
Locate the focus and directrix
Why: Three units either way.
\[ (0, 3)\text{ and } y = -3 \]
Figure (svg): The solution to Worked example the familiar orientation shown as a ladder of expressions, one row per legal move
\[ \text{focus }(0,3),\; \text{directrix }y=-3 \]
Verify: compare with the chapter 3 form
Why: Solving for the vertical variable gives it as the square over twelve, which is the familiar quadratic form with a small leading coefficient. The two descriptions are the same curve, and the focal one adds the focus and directrix that the quadratic form never mentioned.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1238-1239
Error analysis
A student reads a parabola's orientation.
Annotate
On: \( y^2=8x \;\Longrightarrow\; \text{opens vertically, since }y\text{ is squared} \)
The squared variable takes both signs for each value of the other, which is symmetry across the axis. The unsquared one is the direction the curve travels, which is why it names the opening.
Faded example
From an equation with the constant 12.
Fill in the blanks
4p=12 \;\Longrightarrow\; p=3, \text3___\text___
Why: The standard form writes the constant as four times the focal distance, so dividing by four recovers it. That one number locates both the focus and the directrix.
Sorting
The unsquared variable and the sign decide.
Sort into buckets
Sort each equation.
Step zero
You are given a parabola's equation to describe.
Discussion prompt
What two things do you read off?
Hint: One names the axis and one the direction.
Answer:
Which variable is squared, which names the axis of opening — the curve opens along the other one.
And the sign of the constant, which decides which of the two directions along that axis.
Those two readings give one of four orientations, after which only the focal distance remains. Both are visible without any computation, which is why they come first.
Section
Section 3
Concept
The focal distance locates the focus on one side of the vertex and the directrix the same distance on the other. The axis runs through both, perpendicular to the directrix.
The latus rectum — the chord through the focus perpendicular to the axis — gives two more points on the curve for free, which makes a sketch far more accurate than the vertex alone. Its half-length is twice the focal distance in each direction.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1239-1243
Picture it
One constant sets all the distances.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
Everything is measured from the vertex: the focus one way, the directrix the other, and the curve opening towards the focus. There is no second constant to keep track of.
Worked example
From the constant alone.
\[ \text{Find the focus, directrix and axis of } y^2=16x. \]
Find the focal distance
Why: Divide the constant by four.
\[ p = 4 \]
Locate the focus
Why: Four units in the opening direction.
\[ (4, 0) \]
Locate the directrix
Why: Four units the other way.
\[ x = -4 \]
State the axis
Why: Through vertex and focus.
Figure (svg): A parabola with its focus, directrix and vertex marked, and a point on the curve joined to the focus and dropped perpendicular to the directrix
\[ (4,0),\; x=-4,\; y=0 \]
Verify: check the vertex is equidistant
Why: The vertex at the origin is 4 from the focus and 4 from the directrix — equal, as the definition requires for every point including this one. The focus and directrix are always symmetric about the vertex.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1240-1242
Prediction
A parabola opens rightward with focus at four.
Predict first
Where is the directrix?
Correct: A vertical line four units left of the vertex.
Why: The focus and directrix straddle the vertex at equal distances, and the curve opens towards the focus. The directrix is perpendicular to the axis, which is horizontal here, so the line is vertical.
Worked example
Two more points, for free.
\[ \text{Find two more points on } y^2=16x \text{ using the latus rectum.} \]
Recall its length
Why: Four times the focal distance.
\[ 16 \]
Halve it
Why: Each side of the axis.
\[ 8 \]
Go to the focus
Why: Then eight each way.
\[ \text{at } x = 4 \]
Write the points
Why: Above and below.
\[ (4, 8)\text{ and } (4, -8) \]
Figure (svg): The solution to Worked example use the latus rectum shown as a ladder of expressions, one row per legal move
\[ (4,8)\text{ and }(4,-8) \]
Verify: check in the equation
Why: Sixty-four equals sixteen times four — correct for both points, since squaring removes the sign. Those two points plus the vertex give a sketch far more accurate than the vertex alone, at the cost of one division.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1242-1243
Trap
\[ y^2=16x \;\Longrightarrow\; \text{focus }(4,0), \text{ directrix }x=4 \]
Place both at the same distance in the same direction
Why: The directrix is put on the focus's side of the vertex.
The vertex would then be four from the focus and zero from the directrix, which is not equidistant.
The focus and directrix are on opposite sides of the vertex. The curve opens towards the focus and away from the directrix.
They are the same distance from the vertex, which is what makes the vertex equidistant and therefore on the curve.
Checking the vertex is the test: its two distances must be equal, and they are only if the focus and directrix straddle it.
Faded example
With a focal distance of 4.
Fill in the blanks
\text4=4p=4(16)=___
Why: The chord through the focus perpendicular to the axis has length four times the focal distance, so half of it extends each way. Those two endpoints are quick extra points for a sketch.
Sorting
The focus and directrix straddle it.
Sort into buckets
Sort each feature for a rightward-opening parabola.
Explain it
The other conics needed two constants and this needs one.
Discussion prompt
Explain to a classmate why.
Hint: How much freedom does the shape have?
Answer:
An ellipse has two independent lengths — how long and how wide — so it needs two constants. Changing one without the other genuinely changes the shape.
A parabola has only a scale: every parabola is a scaled copy of every other. Stretching one uniformly gives any other.
So one constant, the focal distance, fixes it completely. A good explanation notes the consequence: all parabolas are similar figures, where ellipses of different eccentricities are not.
Section
Section 4
Concept
Replacing each variable by its difference from the vertex's corresponding coordinate translates the whole configuration, with the focus and directrix moving with it.
Completing the square is easier here than for the other conics because only one variable is squared. The other appears to the first power and is simply isolated on the far side.
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1243-1246
Picture it
Translation preserves whichever applies.
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
Moving the vertex changes no orientation. The squared variable and the sign of the constant are untouched by a shift.
Worked example
Read the vertex, then measure from it.
\[ \text{Describe } (y-2)^2=8(x+1). \]
Read the vertex
Why: Opposite signs to those shown.
\[ (-1, 2) \]
Find the squared variable
Why: The vertical one.
Read the focal distance
Why: Constant over four.
\[ p = 2 \]
Place the focus and directrix
Why: Two units either way.
\[ (1, 2)\text{ and } x = -3 \]
Figure (svg): Four small parabolas opening right, left, up and down, each labelled with the sign and variable that produces it
\[ \text{vertex }(-1,2),\; \text{focus }(1,2),\; x=-3 \]
Verify: check the vertex is equidistant
Why: From the vertex to the focus is 2 units and to the directrix is also 2 — equal, so the vertex is on the curve as it must be. The focus and directrix straddle it, both having moved with the translation.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1244-1245
Prediction
An expanded parabola equation.
Predict first
How many squares need completing?
Correct: One, since only one variable is squared.
Why: A parabola's equation has exactly one squared variable; the other appears to the first power and is isolated. That is what distinguishes it from an ellipse or hyperbola, both of which have two squared terms.
Worked example
Only one variable is squared.
\[ \text{Put } y^2-6y-4x+13=0 \text{ into standard form.} \]
Group the squared variable
Why: Leave the other alone.
\[ y ^{2} - 6 y = 4 x - 13 \]
Complete the square
Why: Half of six, squared.
\[ \text{add } 9\text{ to both sides} \]
Factor the left
Why: A perfect square.
\[ (y - 3) ^{2} = 4 x - 4 \]
Factor the right
Why: To reveal the vertex.
\[ (y - 3) ^{2} = 4(x - 1) \]
Figure (svg): The solution to Worked example complete the square shown as a ladder of expressions, one row per legal move
\[ (y-3)^2=4(x-1) \]
Verify: check the vertex
Why: Substituting the vertex makes the left side zero and the right side zero — consistent, and the vertex is on the curve. The focal distance is one, so the focus is at (2, 3) and the directrix is the line at x equal to zero.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1245-1246
Error analysis
A student prepares a parabola's equation.
Annotate
On: \( y^2-6y-4x+13=0 \;\Longrightarrow\; \text{complete the square in }x\text{ too} \)
The presence of exactly one squared variable is what distinguishes a parabola from the other conics on sight. It also makes the algebra shorter, since only one square needs completing.
Faded example
From an equation showing y minus 2 and x plus 1.
Fill in the blanks
\text-1=(2,\;___)
Why: The standard form subtracts the vertex's coordinates, so the signs displayed are the opposite of the vertex's. Substituting the vertex should make both sides zero, which is the check.
Sorting
The count identifies the conic.
Sort into buckets
Sort each equation type.
Explain it to yourself
The number of squared terms names the conic.
Discussion prompt
Explain how to identify a conic from an expanded equation.
Hint: Count and compare.
Answer:
Count the squared terms. One squared variable means a parabola; two means an ellipse or a hyperbola.
If there are two, look at the sign between them: the same sign gives an ellipse and opposite signs a hyperbola.
Equal coefficients on two same-signed squares gives a circle, the ellipse's special case. A good explanation notes that this classification happens before any completing of squares, so it tells you what you are working towards.
Section
Section 5
Concept
Any ray travelling parallel to the axis reflects off the parabola directly to the focus, and a source at the focus produces a parallel beam.
A spherical mirror does not have this property — parallel rays reflect to slightly different points, which is spherical aberration. The parabola's shape is the one that focuses them all exactly, which is why precision instruments use it.
Figure (svg): A parabola with parallel incoming rays reflecting off the curve and converging on the focus
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1246-1247
Picture it
Parallel in, all to one point.
Figure (svg): A parabola with parallel incoming rays reflecting off the curve and converging on the focus
Running the picture backwards gives the other use: a bulb at the focus sends out a parallel beam, which is what a headlight reflector does.
Worked example
The focus is where the signal collects.
\[ \text{A dish has equation } x^2=20y \text{ in cross-section. Where does the receiver go?} \]
Find the focal distance
Why: Constant over four.
\[ p = 5 \]
Identify the direction
Why: The dish opens upward.
Place the focus
Why: Five units up.
\[ (0, 5) \]
Interpret
Why: The receiver goes there.
\[ 5\text{ units from the base} \]
Figure (svg): A parabola with parallel incoming rays reflecting off the curve and converging on the focus
\[ (0,5) \]
Verify: check with the latus rectum
Why: The chord through the focus has length four times five, which is 20 — so the dish is 20 units wide at the height of the receiver. That gives a sense of scale and confirms the focal distance is right for the dish's proportions.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1246-1247
Prediction
Parallel rays strike a parabolic mirror.
Predict first
Where do they meet?
Correct: At the focus.
Why: The reflection property sends every ray parallel to the axis through the focus, exactly. That is why detectors and bulbs are placed there rather than on the mirror's surface.
Worked example
The shape matters.
\[ \text{Why are telescope mirrors parabolic rather than spherical?} \]
Consider a spherical mirror
Why: Rays near the edge behave differently.
Name the effect
Why: The image blurs.
Consider a parabola
Why: The definition forces equal path lengths.
Conclude
Why: Only the parabola focuses exactly.
Figure (svg): The solution to Worked example why not a sphere shown as a ladder of expressions, one row per legal move
\[ \text{no spherical aberration} \]
Verify: connect to the definition
Why: The equal-distance condition means every path from the directrix to the focus via the curve has the same length, so all the reflected waves arrive in step. A sphere has no such property, and its rays arrive slightly out of step — which is the blur.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1247-1247
Trap
\[ \text{the dish is deepest at the vertex, so put the receiver there} \]
Choose the geometric centre of the dish
Why: The deepest point is assumed to be where the signal collects.
Almost none of the reflected signal passes through the vertex.
The reflected rays converge at the focus, which is out in front of the dish rather than on its surface.
That is why a dish antenna has an arm holding the receiver away from the surface, and why the arm's length matters.
The focal distance sets that length, computed from the dish's cross-sectional equation.
Faded example
From a cross-section with constant 20.
Fill in the blanks
4p=20 \;\Longrightarrow\; p=5, \text5___\text___
Why: Dividing the constant by four gives the focal distance, which is where the receiver goes. It sits out in front of the dish rather than on its surface.
Sorting
The property works both ways.
Sort into buckets
Sort each application.
Explain it
It follows from the definition rather than being extra.
Discussion prompt
Explain to a classmate why parallel rays converge.
Hint: What is equal for every point on the curve?
Answer:
Every point of the curve is equally far from the focus and the directrix. So a ray coming in parallel to the axis, which is effectively coming from the directrix, has the same distance still to travel to the focus.
That means every reflected path has the same total length, so all the waves arrive at the focus in step and reinforce.
A sphere has no such property and its rays arrive slightly out of step, which is the blur called spherical aberration. A good explanation notes that the useful physical property is a direct consequence of the geometric definition, not an additional fact to learn.
Comparison
Fill the blanks from memory. The parabola is the structural odd one out.
Comparison matrix
| ellipse | hyperbola | parabola | |
|---|---|---|---|
| defined by | two foci, constant sum | two foci, constant difference | a focus and a directrix |
| squared terms | two, same sign | two, opposite signs | one |
| has a centre | yes | yes | no, a vertex instead |
| constants needed | two | two | one |
The second row is the identification rule. Counting squared terms and comparing their signs names the conic before any other work begins.
Pattern
Five steps, and two of them are readings rather than computations.
Steps 3 and 4 are the two independent choices that give the four orientations. Neither requires any arithmetic beyond a division by four.
OpenStax Algebra and Trigonometry 2e, §12.3 The Parabola §12.3
Check
The definition.
Check your understanding
What is equal for every point on a parabola?
Answer: A
Why: A parabola is defined by one point and one line, with every point on the curve equidistant from them. Using two foci would define an ellipse or a hyperbola instead.
Check
Orientation.
Check your understanding
In an equation where the vertical variable is squared, which way does the parabola open?
Answer: A
Why: The squared variable is symmetric about the axis rather than along it, so the curve opens along the other axis. The sign of the constant then decides left or right.
Check
The reflection property.
Check your understanding
Where should a receiver be placed in a parabolic dish?
Answer: A
Why: Parallel rays reflect through the focus, which lies away from the dish's surface. That is why dish antennas have an arm holding the receiver at exactly the focal distance.
Real world
A car headlight runs the reflection property backwards.
Discussion prompt
Why is a headlight reflector parabolic, and where is the bulb?
Hint: What does the property look like in reverse?
Answer:
A source at the focus sends rays that reflect off the parabola into a parallel beam — the reverse of a dish collecting parallel rays.
So the bulb sits exactly at the focus, and the beam's tightness depends on how precisely it is positioned. A bulb slightly off the focus produces a spreading or converging beam instead.
Which is why headlight aim is a regulated safety matter: a millimetre of bulb misplacement becomes metres of beam displacement at a hundred metres down the road. The geometry that makes the beam possible also makes it sensitive.
Commit first
State your confidence along with your answer.
Predict first
Why does a parabola have no centre?
Correct: It is defined by a point and a line, which have no midpoint between them.
Why: A centre is the midpoint of two foci, and a parabola has one focus and a directrix instead. Having fewer defining objects of the same kind is what leaves it with a vertex rather than a centre.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
A classmate asks how the parabolas of chapter 3 relate to the ones here.
Hint: What is added?
Answer:
They are the same curves. Chapter 3 gave a formula and graphed it; this section gives the definition that formula satisfies.
What is added is the focus and directrix, which the quadratic form never mentioned. Solving the standard form for the unsquared variable recovers exactly a quadratic.
And the definition explains things the formula could not — why parabolic dishes focus and spherical ones do not. A good explanation notes that having a definition rather than just a formula is what makes the physical property derivable.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The second is where the standard error lives, since the squared variable names the axis the curve does not open along. The fourth is the section's clearest connection to anything outside mathematics.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw a parabola with its focus, directrix, vertex and axis labelled, and mark one point with its two equal distances. Beside it, draw the four orientations with the equation form under each. Underneath, complete the square on one expanded equation and place all the parts. Finish with a sketch of the reflection property.
If your four orientations show the squared variable naming the axis the curve does not open along, the section's one recurring error is accounted for.
Recap
Five things, and the second is the one worth checking every time.
| if you remember one thing | it should be this |
|---|---|
| about the definition | equal distances to a point and a line |
| about orientation | the curve opens along the axis of the UNsquared variable |
| about the parts | focus and directrix straddle the vertex at equal distances |
| about applications | the focus is out in front, never on the surface |
Section 10.4 handles conics that have been rotated, where a cross term appears in the equation and the classification rules of this chapter need a new invariant to survive it.
OpenStax, Precalculus, §10.3 The Parabola §10.3, pp. 1230-1247 — everything on these slides traces back here
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