Algebra I · Unit 6: Quadratics & Data · Elise's Elected Education

You throw a basketball straight up. It goes up… up… reaches a peak… then comes back down.

If you tracked its height every second, the numbers wouldn't be a straight line. They'd go: low, higher, highest, lower, low — a curve. That curve is called a parabola, and it's the shape of every quadratic.

A quadratic equation has an x² term. Like:

y = -16x² + 32x + 5

(That's a real ball-toss formula — gravity, initial throw speed, height of your hand.)

The simplest quadratic is x² = 49. It's asking: "what number, multiplied by itself, gives 49?"

Answer: 7 (because 7·7 = 49). But also -7 (because -7·-7 = 49 too — two negatives make a positive).

x = ±7

That little ± ("plus or minus") is huge. Quadratics almost always have two answers. That makes sense for a ball: it passes the same height TWICE — once going up, once coming down.

This unit also covers data — how to read scatterplots and trend lines. Same idea: looking at numbers and finding the pattern in them.

Every quadratic graphs as a U-shaped curve called a parabola.

  y
  ↑
  |  \     /
  |   \   /
  |    \_/ ← VERTEX (the bottom of the U)
  |
  |————————→ x

parabolas open up (smile) or down (frown)

Key parts of a parabola:

Vertex — the very top or bottom point. The "peak" or "valley."

Axis of symmetry — the vertical line through the vertex. Whatever happens on one side mirrors on the other.

x-intercepts (zeros / roots) — where the curve crosses the x-axis. These are the SOLUTIONS to the quadratic.

For y = x² − 4x + 3, the vertex sits at x = 2. You can find it from the formula:

x_vertex = -b / (2a)

Here a = 1 and b = -4, so x = -(-4) / (2·1) = 4/2 = 2. Then plug x=2 back: y = 4 - 8 + 3 = -1. Vertex = (2, -1).

Scatterplots and trend lines. When you have a bunch of data points and you draw a line that best fits the cloud — that's a trend line. If the line slopes UP, the two things are positively related (more X → more Y). If it slopes DOWN, they're negatively related. If it's flat or random, no real relationship.

Three ways to solve a quadratic. Pick whichever fits the problem in front of you.

METHOD 1 — Square root. For equations like x² = 49 with no middle term.

x² = 49 → x = ±√49 = ±7

Don't forget the ± — there are always TWO answers (positive and negative).

METHOD 2 — Factoring. Same trick as Unit 5: find two numbers that multiply to the constant and add to the middle coefficient.

x² + 7x + 10 = 0
(x + 2)(x + 5) = 0
x = -2 or x = -5

Once it's factored, set each factor equal to 0 and solve. (If A·B = 0, then either A=0 or B=0.)

METHOD 3 — Quadratic formula. For any quadratic ax² + bx + c = 0, no matter how messy:

x = (-b ± √(b² − 4ac)) / (2a)

Plug in a, b, c. The ± gives you both solutions. This is the universal tool — it works on everything.

Difference of squares special. If you see x² − 9, that factors as (x + 3)(x − 3). Same with x² − 25 = (x + 5)(x − 5). The pattern: a² − b² = (a + b)(a − b).

Rule of thumb: If there's no middle term, use square root. If it factors easily, factor it. If it's a mess, quadratic formula. Always check by plugging answers back in.

Quadratics & Data: When the Shape Curves

Pick how you want me to explain

Try a few

🌟 Show what you've got

tutor parabolas, factoring, data

Pause. Anytime. Forever if you want.