🎯⭐ INTERACTIVE LESSON

Parabolas (Conic Form)

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Parabolas (Conic Form) - Complete Interactive Lesson

Part 1: Focus, Directrix & the Definition

🛰️ Parabolas (Conic Form)

Part 1 of 5 — Focus, Directrix & the Definition

Topics in This Part

Section
A Parabola as a Conic Section
The Focus–Directrix Definition
The Distance Equation

🔑 Key Concept: In Algebra 1 a parabola was just the graph of $y = ax^2 + bx + c$ . As a conic section, a parabola is defined by a single point (the focus) and a single line (the directrix). Every point on the curve is equidistant from them.

A Parabola as a Conic Section

Slice a cone with a flat plane and you get the four conic sections: a circle, an ellipse, a parabola, and a hyperbola. A parabola is the slice you get when the cutting plane is parallel to one side of the cone.

That geometric origin gives the parabola a special structure that the formula $y = ax^2$ hides. Two features run the whole show:

Feature	What it is
Focus	A fixed point inside the curve
Directrix	A fixed line outside the curve
Vertex	The point halfway between focus and directrix
Axis of symmetry	The line through the focus, perpendicular to the directrix

The Focus–Directrix Definition

🔑 Definition: A parabola is the set of all points that are the same distance from a fixed point (the focus) as from a fixed line (the directrix).

Pick any point $P$ on the parabola. Then:

$\text{distance}(P, \text{focus}) = \text{distance}(P, \text{directrix})$

The vertex sits exactly halfway between the focus and the directrix — so if the focus is units from the vertex, the directrix is also units from the vertex, on the side.

Concept Check 🎯

Turning the Definition Into an Equation

To find a parabola's equation we set the two distances equal. Suppose the focus is $(0, p)$ and the directrix is the line $y = -p$ . Take a general point $(x, y)$ on the curve.

Distance to the focus (distance formula):

Find the Vertex 🧮

The vertex is the midpoint of the focus and the directrix. Find the vertex's $y$ -coordinate for each setup.

1) Focus $(0, 4)$ , directrix $y = -4 \;\Rightarrow\;$ vertex $y = ?$ Focus , directrix vertex Focus , directrix vertex

Name the Parts 🔽

Match each description to the correct piece of a parabola.

Part 2: Vertical Parabolas: x² = 4py

🛰️ Parabolas (Conic Form)

Part 2 of 5 — Vertical Parabolas: $x^2 = 4py$

🔑 The Idea: When the vertex is at the origin and the parabola opens up or down, its equation is $x^2 = 4py$ . The single number controls the focus, the directrix, and which way it opens.

Part 3: Horizontal Parabolas: y² = 4px

🛰️ Parabolas (Conic Form)

Part 3 of 5 — Horizontal Parabolas: $y^2 = 4px$

🔑 The Idea: Swap the roles of $x$ and $y$ and the parabola opens sideways. The equation describes a parabola opening left or right — a function, since it fails the vertical line test.

Part 4: Translated Parabolas (Vertex (h, k))

🛰️ Parabolas (Conic Form)

Part 4 of 5 — Translated Parabolas (Vertex $(h, k)$ )

🔑 The Idea: Slide the whole parabola so its vertex sits at $(h, k)$ . Replace $x$ with $(x - h)$ and with — every feature shifts by the same amount.

Part 5: Mixed Practice & Mastery Check

🛰️ Parabolas (Conic Form)

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) state the focus–directrix definition, (2) read $x^2 = 4py$ , (3) read $y^2 = 4px$ , and (4) handle a translated vertex . Let's put it all together.

💡 Satellite dishes, car headlights, and telescope mirrors are parabolic because rays parallel to the axis all bounce through the focus — that single point is where the signal concentrates.

Distance from vertex	To the focus	To the directrix
Always equal	$p$	$p$

⚠️ The focus is always inside the curve (on the concave side); the directrix is always outside. The parabola bends around its focus and away from its directrix.

(x - 0)^{2} + (y - p)^{2}

Distance to the directrix (vertical distance to $y = -p$ ): $|\,y - (-p)\,| = |\,y + p\,|$

Set them equal and square both sides: $x^2 + (y - p)^2 = (y + p)^2$

Expand and simplify — the $y^2$ and $p^2$ cancel: $x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2 \;\Rightarrow\; x^2 = 4py$

🔑 The master equation $x^2 = 4py$ falls straight out of "equidistant." Part 2 unpacks everything it tells us.

y = 2 \;\Rightarrow\;

y = 1 \;\Rightarrow\;

Reading $x^2 = 4py$

For a parabola with vertex at the origin that opens vertically:

$x^2 = 4py$

Feature	Value
Vertex	$(0, 0)$
Focus	$(0, p)$
Directrix	$y = -p$
Axis of symmetry	the -axis ()

The value $p$ is the focal distance — how far the focus (and the directrix) sit from the vertex.

💡 Don't confuse $p$ with $4p$ . The whole coefficient is $4p$ . To get $p$ , divide the coefficient by $4$ .

Example: $x^2 = 8y$

Here $4p = 8$ , so $p = 2$ .

Opens up (since $p = 2 > 0$ )
Focus $(0, 2)$
Directrix $y = -2$

Solve for $p$ 🧮

Each parabola is $x^2 = 4py$ . Find $p$ by dividing the coefficient by $4$ .

1) $x^2 = 12y \;\Rightarrow\; p = \,?$ 2) $x^2 = 4y \;\Rightarrow\; p = \,?$ 3) $x^2 = -20y \;\Rightarrow\; p = \,?$

Worked Example: Full Description

Describe $x^2 = -16y$ .

Match the form: $4p = -16$ , so $p = -4$ .
Sign of $p$ : negative → opens down.
Focus $(0, p) = (0, -4)$ .
Directrix $y = -p = -(-4) = 4$ , i.e. $y = 4$ .
Axis of symmetry: the $y$ -axis, $x = 0$ .

✅ Sanity check: It opens down, so the focus $(0,-4)$ is below the vertex and the directrix $y = 4$ is above it. The curve wraps around the focus. ✓

Building an Equation From Features

If you know $p$ , just plug into $x^2 = 4py$ .

Vertex $(0,0)$ , focus $(0, 5)$ : the focus is at $(0, p)$ , so $p = 5$ and the equation is .

Describe $x^2 = 20y$ 🔽

Fill in each feature.

Build the Equation 🎯

Reading $y^2 = 4px$

For a parabola with vertex at the origin that opens horizontally:

$y^2 = 4px$

Feature	Value
Vertex	$(0, 0)$
Focus	$(p, 0)$
Directrix	$x = -p$
Axis of symmetry	the -axis ()

Notice the swap from Part 2: the squared variable tells you the axis.

$x^2 = \dots$ → $x$ is squared → opens up/down (vertical axis).
$y^2 = \dots$ → is squared → opens (horizontal axis).

💡 Rule of thumb: The parabola opens toward the variable that is not squared.

Example: $y^2 = 12x$

Here $4p = 12$ , so $p = 3$ .

Opens right (since $p = 3 > 0$ )
Focus $(3, 0)$
Directrix $x = -3$

Which Way Does It Open? 🎯

Worked Example: $y^2 = -20x$

$y$ is squared → horizontal axis.
$4p = -20$ , so $p = -5$ .
$p < 0$ → opens left.
Focus $(p, 0) = (-5, 0)$ .
Directrix $x = -p = -(-5) = 5$ , i.e. $x = 5$ .

✅ Sanity check: Opening left, the focus $(-5,0)$ sits to the left of the vertex and the directrix $x = 5$ is to the right. ✓

Building an Equation From the Focus

Vertex $(0,0)$ , focus $(-4, 0)$ : the focus is $(p, 0)$ , so $p = -4$ . Then

Describe $y^2 = 24x$ 🔽

Fill in each feature.

Find the Focus 🧮

Each parabola is $y^2 = 4px$ with vertex at the origin. Enter the $x$ -coordinate of the focus (which is just $p$ ).

1) $y^2 = 8x \;\Rightarrow\;$ focus $x = \,?$ 2) $y^2 = -12x \;\Rightarrow\;$ focus $x = \,?$ 3) $y^2 = 2x \;\Rightarrow\;$ focus $x = \,?$ (decimal is fine)

The Translated Standard Forms

Opens	Equation	Vertex	Focus	Directrix
Up / Down	$(x - h)^2 = 4p(y - k)$	$(h, k)$	$(h,\, k + p)$	$y = k - p$
Left / Right	$(y - k)^2 = 4p(x - h)$	$(h, k)$

Everything from Parts 2–3 still holds — the focus and directrix just start from $(h, k)$ instead of the origin.

⚠️ Watch the signs. $(x - h)$ means the vertex $x$ -coordinate is $+h$ . So $(x - 3)^2 = \dots$ has , but has .

Example: $(x - 2)^2 = 8(y + 1)$

$x$ squared → opens vertically.
Vertex: $(h, k) = (2, -1)$ (since $y + 1 = y - (-1)$ ).

Read the Vertex 🎯

Worked Example: Full Description

Describe $(y + 2)^2 = 4(x - 3)$ .

$y$ squared → opens horizontally; form is $(y-k)^2 = 4p(x-h)$ .
Vertex: $y + 2 = y - (-2) \Rightarrow k = -2$ ; $x - 3 \Rightarrow h = 3$ . So .
$4p = 4 \Rightarrow p = 1 > 0$ → opens right.
Focus $(h + p, k) = (3 + 1,\, -2) = (4, -2)$ .
Directrix $x = h - p = 3 - 1 = 2$ .

✅ Check: Opens right, so the focus $(4, -2)$ is right of the vertex $(3,-2)$ and the directrix $x = 2$ is left of it. ✓

Building a Translated Equation

Vertex $(1, 5)$ , focus $(1, 8)$ : focus is directly above the vertex → opens up, $h = 1$ , $k = 5$ . The focus distance is . Plug in:

Describe $(x - 4)^2 = 8(y - 1)$ 🔽

Fill in each feature.

Find the Focus Coordinates 🧮

For $(y - 1)^2 = -8(x - 2)$ :

1) Vertex $x$ -coordinate $= \,?$ 2) Vertex $y$ -coordinate $= \,?$ 3) Focus $x$ -coordinate $= \,?$

Quick Reference

Opens	Standard form	Vertex	Focus	Directrix
Up/Down	$(x-h)^2 = 4p(y-k)$	$(h,k)$	$(h, k+p)$	$y = k - p$
Left/Right	$(y-k)^2 = 4p(x-h)$	$(h,k)$

The three-question routine for any parabola:

Which variable is squared? $x^2 \Rightarrow$ up/down; $y^2 \Rightarrow$ left/right.
What is $p$ ? Set the coefficient equal to $4p$ and divide by .

⚠️ Two classic traps: forgetting to divide by $4$ to get $p$ , and flipping the sign of $h$ or $k$ from $(x - h)$ , .

Mixed Practice 🎯

Build & Decode 🧮

1) Vertex $(0,0)$ , focus $(0, 7)$ . In $x^2 = 4py$ , what is the value of $4p$ (the coefficient of $y$ )? 2) For $(x - 2)^2 = 16(y + 3)$ , what is $p$ ? 3) For that same parabola, what is the $y$ -coordinate of the vertex?

Before the Exit Quiz: Three Pitfalls

Pitfall	Fix
Using the coefficient as $p$	The coefficient is $4p$ — divide by 4.
Flipping the vertex sign	$(x - h)$ , $(y - k)$ : the vertex is $+h$ , $+k$ , so $(x+5)$ gives $h = -5$ .
Mixing up which way it opens	Squared variable sets the axis; the sign of $p$ sets the direction.

💡 One last gut-check on any answer: the focus must sit inside the curve and the directrix outside, on opposite sides of the vertex. If yours don't, recheck the sign of $p$ .

Decode This Parabola 🔽

Given $(y + 1)^2 = -12(x - 4)$ , fill in each feature.

Exit Quiz ✅

Answer all three to finish the lesson.

y^2 = 4px = 4(-4)x = -16x.

(x + 3)^2 = (x - (-3))^2

4p = 8 \Rightarrow p = 2 > 0

→ opens up.

Focus

(h, k + p) = (2,\, -1 + 2) = (2, 1)

Directrix

y = k - p = -1 - 2 = -3

vertex $(3, -2)$

(x - 1)^2 = 4(3)(y - 5) = 12(y - 5).

Which sign?

p > 0

opens up/right;

p < 0

opens down/left.

Parabolas (Conic Form)

Parabolas (Conic Form) - Complete Interactive Lesson

Part 1: Focus, Directrix & the Definition

🛰️ Parabolas (Conic Form)

Topics in This Part

A Parabola as a Conic Section

The Focus–Directrix Definition

Turning the Definition Into an Equation

Part 2: Vertical Parabolas: x² = 4py

🛰️ Parabolas (Conic Form)

Part 3: Horizontal Parabolas: y² = 4px

🛰️ Parabolas (Conic Form)

Part 4: Translated Parabolas (Vertex (h, k))

🛰️ Parabolas (Conic Form)

Part 5: Mixed Practice & Mastery Check

🛰️ Parabolas (Conic Form)

Reading x2=4pyx^2 = 4pyx2=4py

Example: x2=8yx^2 = 8yx2=8y

Worked Example: Full Description

Building an Equation From Features

Reading y2=4pxy^2 = 4pxy2=4px

Example: y2=12xy^2 = 12xy2=12x

Worked Example: y2=−20xy^2 = -20xy2=−20x

Building an Equation From the Focus

The Translated Standard Forms

Example: (x−2)2=8(y+1)(x - 2)^2 = 8(y + 1)(x−2)2=8(y+1)

Worked Example: Full Description

Building a Translated Equation

Quick Reference

Before the Exit Quiz: Three Pitfalls

Reading $x^2 = 4py$

Example: $x^2 = 8y$

Reading $y^2 = 4px$

Example: $y^2 = 12x$

Worked Example: $y^2 = -20x$

Example: $(x - 2)^2 = 8(y + 1)$