Which way a parabola opens, and the three ways it can meet that axis

What to assume they know

• A zero is exactly a point where the curve meets the horizontal axis — that a zero of a polynomial and a crossing of the horizontal axis are the same thing said twice
• Degree as the label that separates linear, quadratic and cubic — the general quadratic ax² + bx + c, and why a ≠ 0
• Reading a curve off a grid, including recognising when a curve touches a line without passing through it
• That squaring a large number, positive or negative, gives a large positive number — this is what makes the two ends of the curve behave alike

What they should be able to do

• Predict from the sign of the leading number whether the curve of a quadratic opens upward or downward, and give the reason
• Name the three possible relationships between such a curve and the horizontal axis, and sketch each for both openings
• Read the number of zeroes off a supplied quadratic curve in each of the three cases
• Explain why a curve that touches the axis without crossing is counted as two equal zeroes, while allowing that the chapter itself also calls that case one zero — and say why both descriptions are defensible
• Argue why no fourth case is possible
• State the ceiling this argument establishes for a degree-2 polynomial, and say why it is a ceiling and not a count
• Recognise that a quadratic with no crossing is a legitimate polynomial with legitimate coefficients, not an error

Where it usually goes wrong

• "Every quadratic has two zeroes." The chapter draws two whole figures, Fig. 2.4 and Fig. 2.5, whose entire purpose is to refute this. Two is the maximum, not the outcome.
• "No zero means I made an arithmetic mistake." It means the curve missed the axis. Case (iii) is a picture of a perfectly ordinary quadratic — every coefficient real, a ≠ 0 — that simply has no real input returning 0.
• "A curve that only touches the axis has no zero, because it never crosses." The chapter counts it: the x-coordinate of the touching point is "the only zero" in that case, the place where the two crossing points have come together, which is why the book also calls it "two equal zeroes (i.e., one zero)". Both names are the book's own, and both describe one value.
• "a > 0 means the curve is above the axis." It fixes the direction of opening, and nothing else. Fig. 2.3 panel (ii) and Fig. 2.5 panel (i) both open upward and sit differently against the axis.
• "Whether there are zeroes depends on how far you extend the drawing." For an upward-opening curve the two arms rise forever, so extending the picture cannot bring them back down to the axis. Say this once; it is what makes the three cases exhaustive rather than provisional.
• "There should be a fourth case where the curve runs along the axis." That would need every input to return 0, which no quadratic with a ≠ 0 does.
• "Case (iii) means the polynomial has zeroes we are not allowed to talk about." This chapter works with real numbers throughout, and what it claims is that no real input returns 0. Leave it there — anything further is outside the book.

Questions to check understanding

• Given a supplied parabola, state the number of zeroes and which of the three cases it shows
• Given the sign of the leading number, state which way the curve opens
• Sketch a curve to order: an upward-opening quadratic with no zero, a downward-opening one with a single touch
• Explain why a quadratic cannot meet the horizontal axis at three points
• Decide whether a stated quadratic can have zeroes at all, arguing from the sign of its values rather than from a formula
• Justify, in a sentence, why the three cases exhaust the possibilities

Examples worth working on the board

Statements marked verified were read off the printed pages or worked through by me.

• The two shapes (§2.2, p. 13). The section states the rule linking the sign of the leading number to the direction the curve opens: positive opens one way, negative the other, and it draws both shapes inline in the running text. Verified against the chapter's own worked case: x² − 3x − 4 has leading number 1, which is positive, and the curve of Fig. 2.2 on p. 13 opens upward.
• Fig. 2.3, Case (i) (§2.2, p. 14). Two panels, (i) and (ii), read off the printed page. Panel (i) is a downward-opening curve whose two ends dip below the axis, so it crosses twice; panel (ii) is an upward-opening curve whose lowest part sits below the axis, so it also crosses twice. The two crossings are lettered A and A′ in both panels. Verified: both openings appear, and the count is 2 in each.
• Fig. 2.4, Case (ii) (§2.2, p. 14). Again two panels. Panel (i) opens downward with its highest part resting on the axis; panel (ii) opens upward with its lowest part resting on the axis. A single point is lettered A in each. Verified: the lettering drops from two letters to one, and that drop is the whole content of the case — the chapter describes A and A′ as having come together.
• Fig. 2.5, Case (iii) (§2.2, p. 15). Two panels. Panel (i) is an upward-opening curve sitting entirely above the axis; panel (ii) is a downward-opening curve sitting entirely below it. Verified: neither panel carries an A or an A′, because in this case there is no crossing to letter. The origin O is lettered in both panels, as it is throughout the chapter's graphs; what is missing is any letter marking a meeting with the axis.
• A quadratic that realises Case (iii), to build. The chapter draws the case but does not supply a formula for it in §2.2. Two are available from the chapter's own later data. Exercise 2.2 item 2(iv) asks for a quadratic whose zeroes sum to 1 and multiply to 1; verified: x² − x + 1 fits, and it has no real zero, because it can be rewritten as (x − 1/2)² + 3/4, and a square is never negative, so the value is at least 3/4 at every real input. Item 2(iii) asks for sum 0 and product √5; verified: x² + √5 fits, and by the same reasoning its value is at least √5 everywhere, so its curve lies wholly above the axis. These are the same figures as Fig. 2.5 panel (i), now with an equation attached, and they cost nothing to show.
• The count in each case, stated as a table for the figure. Case (i): two distinct zeroes. Case (ii): two coincident zeroes, one point on the axis. Case (iii): no zero at all. Verified as the conclusion the chapter draws from them: a degree-2 polynomial has at most two zeroes — a ceiling, reached with two distinct zeroes in the first case only.
• The axis lettering used throughout (§2.2, pp. 12–15). Every figure in this chapter letters the horizontal axis X′ to the left and X to the right, the vertical axis Y′ below and Y above, with O at the origin. This lettering sits inside the artwork and is worth reproducing in redrawn figures, since exercise graphs use it too.

Figures to have open

• Fig. 2.3, Fig. 2.4 and Fig. 2.5 redrawn as schematics, each with both of its printed panels and the chapter's own axis lettering. These are the chapter's figures (pp. 14–15) and the argument of this topic is precisely that all six panels are needed; redraw rather than reproduce the printed art.
• A three-by-two comparison grid built from those six panels, so the pattern is visible at once. Standard schematic.
• A movement of the two crossings of Case (i) sliding together into the touch of Case (ii). Not in the book — this is an addition made here, and it is the clearest way to justify counting a touch as two.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 2 "Polynomials", §2.2, pp. 13–15 — the paragraph on the two shapes, then Case (i) with Fig. 2.3, Case (ii) with Fig. 2.4, and Case (iii) with Fig. 2.5, down to the sentence that draws the ceiling of two.
• Forward pointer inside the same chapter: Exercise 2.2 item 2 (p. 23) supplies the sum-and-product data from which the two Case (iii) polynomials above are built.
• The chapter's §2.4 "Summary", p. 23, keeps the ceiling as its fourth point.