Parabolas (Conic Form)

🛰️ Parabolas (Conic Form)

Part 1 of 5 — Focus, Directrix & the Definition

Topics in This Part

🔑 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:

💡 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.

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 $p$ units from the vertex, the directrix is also $p$ units from the vertex, on the opposite side.

⚠️ 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.

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): $\sqrt{(x - 0)^2 + (y - p)^2}$

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.

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 $y = \,?$ 2) Focus $(0, 6)$, directrix $y = 2 \;\Rightarrow\;$ vertex $y = \,?$ 3) Focus $(0, -3)$, directrix $y = 1 \;\Rightarrow\;$ vertex $y = \,?$

Name the Parts 🔽

Match each description to the correct piece of a parabola.

🛰️ 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 $p$ controls the focus, the directrix, and which way it opens.

Reading $x^2 = 4py$

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

$x^2 = 4py$

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 $x^2 = -16y$.

1. Match the form: $4p = -16$, so $p = -4$.
2. Sign of $p$: negative → opens down.
3. Focus $(0, p) = (0, -4)$.
4. Directrix $y = -p = -(-4) = 4$, i.e. $y = 4$.
5. 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 $x^2 = 4(5)y = 20y$.

Describe $x^2 = 20y$ 🔽

Fill in each feature.

Build the Equation 🎯

🛰️ 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 $y^2 = 4px$ describes a parabola opening left or right — not a function, since it fails the vertical line test.

Reading $y^2 = 4px$

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

$y^2 = 4px$

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

• $x^2 = \dots$ → $x$ is squared → opens up/down (vertical axis).
• $y^2 = \dots$ → $y$ is squared → opens left/right (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: $y^2 = -20x$

1. $y$ is squared → horizontal axis.
2. $4p = -20$, so $p = -5$.
3. $p < 0$ → opens left.
4. Focus $(p, 0) = (-5, 0)$.
5. 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 $y^2 = 4px = 4(-4)x = -16x.$

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)

🛰️ 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 $y$ with $(y - k)$ — every feature shifts by the same amount.

The Translated Standard Forms

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 $h = 3$, but $(x + 3)^2 = (x - (-3))^2$ has $h = -3$.

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

• $x$ squared → opens vertically.
• Vertex: $(h, k) = (2, -1)$ (since $y + 1 = y - (-1)$).
• $4p = 8 \Rightarrow p = 2 > 0$ → opens up.
• Focus $(h, k + p) = (2,\, -1 + 2) = (2, 1)$.
• Directrix $y = k - p = -1 - 2 = -3$.

Read the Vertex 🎯

Worked Example: Full Description

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

1. $y$ squared → opens horizontally; form is $(y-k)^2 = 4p(x-h)$.
2. Vertex: $y + 2 = y - (-2) \Rightarrow k = -2$; $x - 3 \Rightarrow h = 3$. So vertex $(3, -2)$.
3. $4p = 4 \Rightarrow p = 1 > 0$ → opens right.
4. Focus $(h + p, k) = (3 + 1,\, -2) = (4, -2)$.
5. 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 $p = 8 - 5 = 3$. Plug in: $(x - 1)^2 = 4(3)(y - 5) = 12(y - 5).$

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 $= \,?$

🛰️ 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 $(h, k)$. Let's put it all together.

Quick Reference

The three-question routine for any parabola:

1. Which variable is squared? $x^2 \Rightarrow$ up/down; $y^2 \Rightarrow$ left/right.
2. What is $p$? Set the coefficient equal to $4p$ and divide by $4$.
3. Which sign? $p > 0$ opens up/right; $p < 0$ opens down/left.

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

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

💡 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.