Algebra 1 · 2026 · V2

The Parabola
Project

A rainbow-shaped journey through one simple equation —
what it means, where it lives, and how to draw it.

By James Algebra 1 2026
scroll to begin ↓
✨
✨
✦ A Surprise ✦

Dear Teacher,

My Parabola Project turned out to be really fun to work on — so I wanted to do something extra. This is my whole poster, brought to life on the web. Every part of the rubric is in here: the direction, the axis of symmetry, the vertex, the max, both intercepts, three other points (one by substitution, two by symmetry), the graph, and my real-life example.

Scroll through it — the rainbow draws itself, the graph is interactive (try hovering the curve), and at the very bottom there's a checklist that maps every rubric item to where it's answered. I hope you enjoy it.

— Thank you for an awesome year,

James

Real-Life Example

My parabola in the real world: a rainbow.

A rainbow is a downward-opening parabola you can see with your own eyes — a smooth arch of color over a huge span of sky.

Where it's found: everywhere on Earth — any open sky where sunlight hits falling raindrops from behind the viewer. Most commonly seen after rain showers, near waterfalls, or in the mist of a garden hose on a sunny day.

Two facts about rainbows

01

Rainbows are full circles.

You never see the whole thing — the horizon (or the atmosphere around you) blocks the bottom half. From a plane or a tall enough mountain, a full-circle rainbow is actually possible.

02

No two people see the same rainbow.

Rainbows are optical illusions that depend on the exact angle between you, the Sun, and the raindrops. Stand one step to the side, and every drop making your rainbow is replaced by a different one.

The Equation

Meet the equation.

This is the parabola I chose for the project. Every single thing on this page comes from these three coefficients.

y = −x2 + 10x + 10
a = −1 b = 10 c = 10
Step 1

Which way does it open?

A parabola opens up or down based on the sign of a — the number in front of x².

  • In our equation, a = −1, which is negative.
  • Negative a means the parabola opens downward.
  • That's the rainbow shape — a smooth arch from low to high to low.
“The parabola for this equation opens downward because the value of a is −1, which is negative.”
opens ↓ downward
because a < 0
Step 2

The axis of symmetry.

Every parabola has a vertical line that cuts it into two mirror-image halves. To find it, use the formula x = −b / (2a).

Formula:  x = −b / (2a)
Plug in:  x = −(10) / (2 · −1)
Simplify:  x = −10 / −2
Answer:  x = 5
“The axis of symmetry is x = 5.”
Step 3

The vertex.

The vertex is the highest (or lowest) point on the parabola — the tip of the rainbow. To get its y-value, plug the axis of symmetry back into the original equation.

Start with:  y = −x² + 10x + 10
Sub x = 5:  y = −(5)² + 10(5) + 10
Compute:  y = −25 + 50 + 10
Vertex:  ( 5 , 35 )
“The vertex is located at (5, 35).”
“The maximum value of this quadratic function is 35 — found by taking the y-coordinate of the vertex, since the parabola opens downward.”
Axis of Symmetry
x = 5
Vertex
(5, 35)
Max / Min
max = 35
Bonus · Two Forms

Same parabola, two forms.

Once you know the vertex, you can rewrite the equation in vertex form: y = a(x − h)² + k, where (h, k) is the vertex. Both forms graph the exact same rainbow.

Standard form
y = −x2 + 10x + 10

Good for spotting the y-intercept (it's just c) and for plugging into the quadratic formula.

Vertex form
y = −(x − 5)2 + 35

The vertex (5, 35) pops right out of the equation — h = 5, k = 35. Perfect for seeing the maximum at a glance.

Why they're the same equation

Expand vertex form:  y = −(x − 5)² + 35
FOIL the square:  y = −(x² − 10x + 25) + 35
Distribute −1:  y = −x² + 10x − 25 + 35
Combine:  y = −x² + 10x + 10  ✓ back to standard form
Step 4

Where it crosses the axes.

The y-intercept is where the parabola crosses the y-axis (when x = 0). The x-intercepts are where it crosses the x-axis (when y = 0).

Y-intercept

To find the y-intercept, plug x = 0 into the equation:

y = −(0)² + 10(0) + 10
y = 10
“The y-intercept for this equation is (0, 10).”

X-intercepts

Set y = 0 and solve −x² + 10x + 10 = 0 with the quadratic formula x = (−b ± √(b² − 4ac)) / (2a).

Discriminant:  b² − 4ac = (10)² − 4(−1)(10) = 100 + 40 = 140
Plug in:  x = (−10 ± √140) / (2 · −1)
Simplify:  x = (−10 ± √140) / −2  =  5 ∓ √35
Answer:  x ≈ −0.92  or  10.92
“The x-intercepts for this equation are (−0.92, 0) and (10.92, 0).”
Y-intercept
(0, 10)
X-intercept (left)
(−0.92, 0)
X-intercept (right)
(10.92, 0)
Step 5 · Three Other Points

Three more points on the curve.

The rubric asks for at least three other points — one found by substitution, the rest found by using the parabola's symmetry across the axis x = 5.

Point 1 — by substitution

Pick x = 3 and plug it into the original equation:

Start with:  y = −x² + 10x + 10
Sub x = 3:  y = −(3)² + 10(3) + 10
Compute:  y = −9 + 30 + 10
Point:  ( 3 , 31 )

Point 2 — by symmetry

Since x = 3 is 2 units left of the axis x = 5, go 2 units right to x = 7. By symmetry, the y-value is the same.

Mirror of x = 3:  5 + (5 − 3) = 7
Point:  ( 7 , 31 )

Point 3 — by symmetry

The y-intercept (0, 10) is 5 units left of the axis. Its mirror is 5 units right, at x = 10, with the same y-value.

Mirror of x = 0:  5 + (5 − 0) = 10
Point:  ( 10 , 10 )
“Three other points on the parabola are (3, 31), (7, 31), and (10, 10). (3, 31) was found by substitution; the other two were found by reflecting across the axis of symmetry x = 5.”
The Graph

The whole rainbow.

Move your mouse (or finger) along the curve — every point on the parabola gives you its exact (x, y) coordinates.

y = −x² + 10x + 10

scale: 1 unit per gridline  ·  hover the curve
Vertex (5, 35) Y-intercept (0, 10) X-intercepts (−0.92, 0) · (10.92, 0) Axis of symmetry (x = 5) Plotted points from the table
Plotting by hand

Finding points, one x at a time.

To plot a parabola, plug different x-values into the equation and write down what y comes out. Because a parabola is symmetric, once you have one side, the other side mirrors across the axis of symmetry.

xy = −x² + 10x + 10point

How to use the table

Every row is a dot on the graph. Here's the trick that saves half the work:

  • Pick an x-value, plug it in, and compute y. That's one point.
  • Find the distance from x to the axis of symmetry (x = 5).
  • The same distance on the other side gives the mirror point — same y, different x.
  • Example: x = 4 gives y = 34. The mirror is x = 6, also y = 34.
Your turn

Try any x-value.

Type a number — any number — and watch it get plugged into the equation, step by step.

What's happening?

You're doing the exact same substitution the project asks for — but live, with any x-value you want. Try big numbers (x = 20) to see how fast it drops off. Try x = 5 to hit the vertex right on.

Reflection

What I actually learned.

A project is just work until you notice the shape of the thing. Here's what clicked for me along the way.

01

The sign of a decides everything.

One tiny minus sign flips the whole rainbow upside down. If a were +1, the parabola would open upward and have a minimum instead of a max — same equation shape, completely different picture.

02

Symmetry is a shortcut.

Once I had the axis at x = 5, I only had to compute half the points — the other half mirror for free. That's why (3, 31) and (7, 31) share a y-value.

03

The vertex form tells the secret.

Writing y = −(x − 5)² + 35 makes the vertex (5, 35) literally appear in the equation. Same parabola, just a form where the answer is already visible.

04

Parabolas are everywhere.

The rainbow example wasn't forced — once I started looking, arches are everywhere: bridges, fountains, thrown balls, the path of a high-five. Math isn't hiding; I just had to look for the shape.

Rubric Check

Everything the project asks for.

Every item from the Algebra 1 Parabola Project 2026 rubric, mapped to where it's answered on this page.

  • ✓
    a. Does the parabola open upward or downward?
    Opens downward because a = −1 < 0.
    Step 1 →
  • ✓
    b. Axis of symmetry
    x = −b/(2a) = 5
    Step 2 →
  • ✓
    c. Vertex coordinates
    (5, 35) — found by plugging x = 5 into the equation.
    Step 3 →
  • ✓
    d. Maximum / minimum
    Maximum = 35 — y-coordinate of the vertex, since the parabola opens downward.
    Step 3 →
  • ✓
    e. Y-intercept
    (0, 10) — plug x = 0 into the equation.
    Step 4 →
  • ✓
    f. X-intercepts
    (−0.92, 0) and (10.92, 0) — via the quadratic formula, discriminant = 140.
    Step 4 →
  • g
    g. There is no g. 🙂
    The teacher already told us: there is no g.
    —
  • ✓
    h. Three other points (1 by substitution, rest by symmetry)
    (3, 31) by substitution, (7, 31) and (10, 10) by symmetry.
    Step 5 →
  • ✓
    i. Graph the parabola with all points labeled
    Interactive graph with scale, vertex, intercepts, and symmetry points labeled.
    Graph →
  • ✓
    j. Real-life example + location + two facts
    Rainbow — seen anywhere sunlight hits falling raindrops. Two facts included.
    Rainbows →
✨   ✨   ✨

Thank you.

Thanks for being the kind of teacher who makes a project about one equation turn into an excuse to draw rainbows and think about how parabolas are actually everywhere. I had a lot of fun with this one.

Hope you enjoyed the rainbow.

— James