No drilling here. The goal is for the ideas to make sense, so you're thinking — not just copying a pattern.
If you understand the WHY, you can solve a problem you've never seen before. That's the real skill.
The whole exam, in one sentence:
You're studying curves called parabolas (anything with an x² in it) — how to read them, find their special points, and solve them — plus how to find several unknown numbers at once from a set of clues (systems).
A "quadratic" is just any expression where the highest power of x is x² (x squared). That little ² changes everything.
The key idea: the power of x tells you the shape.
x (power 1) → a straight line. Example: y = 2x + 1.
x² (power 2) → a curve that bends, shaped like a U. Example: y = x² − 4.
That U-shaped curve has a name: a parabola. (parábola)
y = 2x + 1 → a straight line (power 1)
y = x² → a curved U / parabola (power 2)
Tiny example — feel the difference
Take y = x² and plug in some x values:
x = −2
y = (−2)² = 4
x = −1
y = 1
x = 0
y = 0
x = 1
y = 1
x = 2
y = 4
Notice: −2 and +2 give the same answer (4). That's because squaring kills the minus sign. That's why the curve is a symmetric U — both sides are mirror images. It goes down, hits a bottom, and comes back up.
Where you've seen this in real life 🏀
Throw a ball. It goes up, slows, reaches a highest point, then comes back down. The path it draws in the air is a parabola. A fountain of water, a basketball shot, a jump — all parabolas. Math is just describing that arc with x².
🤔 Think, don't memorize: Why can't a parabola be a straight line? Because of the x² — squaring makes small x's tiny and big x's huge, so the curve gets steeper the further out you go. The bend is built into the x².
② The parabola and its special parts
Every parabola has the same few special points. The exam keeps asking for these same points over and over. Once you know what each one means, you'll never confuse them.
The same curve, with every part the exam asks about labelled. ● zeros · ● vertex · ● y-intercept.
What each part MEANS (not just how to find it)
Opens up or down — does the U smile 😀 (up) or frown ☹️ (down)?
Decided by the number in front of x² (we call it a). Positive a → smile (up), has a lowest point. Negative a → frown (down), has a highest point. (a positivo = abre hacia arriba)
Vertex(vértice) — the turning point. The very bottom of the smile (or top of the frown). It's the most important point: the curve is a mirror around it. Think: the exact moment the ball stops rising and starts falling.
Zeros / roots(raíces / ceros) — the spots where the curve touches the ground (the x-axis). At those spots y = 0. Think: where the ball hits the floor. A parabola can hit the floor twice, once, or never.
Y-intercept(ordenada al origen) — where the curve crosses the vertical line, i.e. when x = 0. Think: the ball's starting height the instant you let go.
Domain(dominio) — all the x-values allowed. For a parabola it's always every number (ℝ), because you can square anything. (Easy free mark.)
Image / Range(imagen) — all the y-values the curve actually reaches. A smile starts at its bottom and goes up forever, so the image is "y ≥ the vertex height". Think: the ball reaches every height from the ground up to its peak — nothing above the peak.
Positivity / Negativity(positividad / negatividad) — positive = where the curve is above the ground (y > 0); negative = where it's below (y < 0). The zeros are the borders between "above" and "below".
🤔 Think: Why are the zeros the borders for positive/negative? Because a zero is exactly where the curve touches the ground — on one side it's above, on the other it's below. The curve can only switch between + and − by passing through 0.
🎮 Parabola Playground — drag the sliders!
This is the general parabola y = a·x² + b·x + c. Move the sliders and watch the curve change in real time. Notice how a opens/flips it, and how the vertex and zeros move. Play until it feels obvious. 🎈
a =1
b =-2
c =-3
③ "Solving an equation" = finding where it equals zero
When a problem says "solve x² + ... = 0", in plain words it's asking: "For which x is this curve sitting on the ground?" You're hunting for the zeros.
The most powerful little rule (Zero Product Property):
If two things multiply to give zero, then at least one of them must be zero. A · B = 0 → A = 0 or B = 0 Si dos cosas multiplicadas dan 0, una de las dos TIENE que ser 0.
Why "factoring" works — tiny example
x(x − 5) = 0
Two things are multiplied: x and (x−5). Their product is 0, so one of them is 0:
x = 0 OR x − 5 = 0 → x = 5
So the curve touches the ground at x = 0 and x = 5. That's the whole trick — break it into a product, then set each piece to zero.
The three tools — and the ONE idea behind them
All three do the same job (find the zeros). You just pick the easiest one for the shape in front of you:
Factoring — turn it into a product, use the zero rule. Best when it factors nicely.
Square-root method — when it looks like (something)² = number, undo the square by rooting both ways (±). Think: "what number squared gives this?" — there are always two (a + and a −).
The Quadratic Formula — the master key. It always works, even when nothing factors. You just feed it a, b, c.
The discriminant = "how many times does it touch the ground?"
Inside the formula there's a part under the square root: b² − 4ac. That number alone tells you the answer count, before you even finish:
Positive → the U crosses the ground twice → 2 solutions.
Zero → the U just kisses the ground at the bottom → 1 solution.
Negative → the U floats and never touches the ground → no real solution. (no se puede sacar raíz de un negativo)
🤔 Think: Why does a negative under the root mean "no solution"? Because no real number squared gives a negative (even × even = +, odd... still +). So if the formula asks for √(−12), there's no real number that fits → the curve simply never reaches the ground.
④ "Analyzing a function" = describing the whole curve
When the exam gives you a function and says "determine domain, image, vertex, zeros…", it's really saying: "Describe this parabola completely, then draw it." You're a detective writing the curve's full ID card.
Think of it like describing a person 🪪
Domain & image = "where they can be". Vertex = "their key feature". Zeros = "where they meet the ground". Y-intercept = "where they start". Positivity/negativity = "when they're up vs down". You're not doing 7 random tasks — you're building one full picture of the same curve, and every piece helps you draw the graph.
Why two "forms" of the same function exist — they're shortcuts that hand you an answer for free:
Factored forma(x − r₁)(x − r₂) → the zeros are sitting right there (r₁ and r₂). Like the equation is already half-solved.
Vertex forma(x − h)² + k → the vertex (h, k) is sitting right there. You can read the turning point with no calculation.
Same curve, written two ways, each revealing a different secret. ¡Ojo con los signos! (x − 3) significa que la raíz/vértice está en +3.
🤔 Think: If you know the two zeros, you already know where the vertex is left-to-right — it's exactly in the middle of them (because the parabola is symmetric). That's why "vertex x = midpoint of the zeros" works. You're not memorizing a formula; you're using the mirror symmetry.
⑤ Systems of equations — finding several unknowns at once
A single equation like x + 2 = 5 has one unknown and one answer. A system gives you several equations that must ALL be true at the same time, to pin down several unknown numbers together.
Think of it as a detective puzzle 🕵️
"I'm thinking of three numbers x, y, z. Clue 1: they add up to 6. Clue 2: x minus y plus z is 2. Clue 3: ..." One clue alone isn't enough — lots of numbers add to 6. But all the clues together point to exactly one trio. Solving the system = combining the clues until only one answer survives.
The core idea: get rid of unknowns one at a time.
Three unknowns feels scary, so you shrink the problem: combine two clues to cancel one letter → now you have a smaller puzzle with 2 unknowns → cancel again → down to 1 unknown you can just solve → then walk back to fill in the others. (Eliminás de a una incógnita hasta que queda una sola.)
Mini example with 2 unknowns (the same idea, smaller)
x + y = 10 and x − y = 4
Add the two equations: the +y and −y cancel → 2x = 14 → x = 7.
Then from the first: 7 + y = 10 → y = 3. Done: x = 7, y = 3.
See the move? We combined the clues to make a letter disappear. With 3 unknowns you do this same trick twice. Con 3 incógnitas, repetís este truco dos veces.
🤔 Think: Why must we check our answer at the end? Because the right trio has to satisfy every clue. If you plug x, y, z back in and one equation doesn't balance, you made a slip somewhere — checking catches it and it's free marks.
✅ Quick self-check — can you explain these in your own words?
If you can answer these out loud (to yourself, or to Dad 😄), you've understood, not memorized. Then go to the Study Guide and Workbook.
What makes an equation "quadratic", and what shape does it draw?
What does the vertex mean? The zeros? The y-intercept?
Why does "A · B = 0" let us solve by factoring?
What does the discriminant being negative tell you about the graph?
Why does knowing the two zeros tell you where the vertex is?
In a system, why isn't one equation enough to find the numbers?
🎮 Faustina's Ready-Check Quiz
Click the answer you think is right for each question. It tells you straight away if you're correct. When all 8 are done, press "See my score".
🎯 You need 6 out of 8 to be ready for the exercises. You can always retake it — getting it wrong is part of learning!
1) Solve: x + 4 = 9. What is x?
2) What is (−4)² ?
3) The graph of y = x² is what shape?
4) The vertex of a parabola is…
5) The zeros (roots) of a function are where…
6) If (x − 2)(x + 5) = 0, the solutions are…
7) If the discriminant b² − 4ac is negative, the equation has…
8) To find the y-intercept of a function, you…
Understand first, then practice. Once these ideas click, the exercises are just the same ideas wearing different numbers. 🧠💜