What a and b do to the line: slope, y-intercept, and parallel families

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What to assume they know

Why two points are enough to draw the line — plotting a linear equation from two points and testing a third by substitution
Recovering y = ax + b from two observations — the form y = ax + b and recovering its two constants
A constant difference is the signature of a linear pattern — the fixed gap between consecutive terms of a linear pattern
Linear growth and linear decay — linear growth and linear decay
The four quadrants, and reading a point's signs off its position — the quadrants and the signs of coordinates
Comparing fractions with 1, to sort coefficients such as ½, 1 and 2
Reading a negative number as a signed position rather than a size

What they should be able to do

Read a and b off an equation written in the form y = ax + b
Predict how a line changes when a alone is altered, and when b alone is altered
Explain why every line of the form y = ax passes through the origin
Compare the steepness of two lines through the origin from their coefficients, including negative coefficients
Relate the sign of a to linear growth and linear decay
Give, as a coordinate pair, where y = ax + b meets the vertical axis, and read the y-intercept off an equation
Decide whether two given lines are parallel by comparing their coefficients
Identify the odd line out in a set that shares no common coefficient
Connect a to the fixed gap of the corresponding numerical pattern

Where it usually goes wrong

"A bigger a always means a steeper line." True for positive coefficients. For negative ones it is the size of a that decides: y = –3x is steeper than y = –x, though –3 is the smaller number. The chapter poses this case and leaves it open.
"A negative slope means the line is below the axis." It means the line falls as you move right. y = –3x + 1 is above the axis for every negative x. Show it crossing.
"Changing b tilts the line." It slides it. Show y = 2x – 1 up to y = 2x + 1 and y = 2x + 5 and keep the direction visibly locked — that is Fig. 2.13's whole content.
"Changing a moves the crossing point." For lines written as y = ax + b it does not: at x = 0 the ax term is zero whatever a is. The chapter states this on p. 36, and the Think and Reflect pair y = 3x + 1 and y = –3x + 1 on p. 33 demonstrates it.
"Parallel lines look parallel, so I can judge by eye." The test is arithmetic: equal a. Fig. 2.14's three lines all lean the same general way and none of them are parallel.
"Every line goes through the origin." Only those with b = 0. The chapter spends Figs. 2.8 to 2.11 on that family and then deliberately leaves it in Figs. 2.12 to 2.14.
"The y-intercept is a distance, so it cannot be negative." It is a signed position on the axis. The chapter's own value of –2 for y = 3x – 2 settles it, and the summary's looser wording on p. 40 should not be followed.
"y = 2x + 3 is parallel to y = –2x." It is not — one climbs and one falls. This is exactly the trap in Exercise Set 2.6 part (v), and a student who assumes every listed group is parallel will get it wrong.
"Slope and constant difference are different ideas that happen to agree." They are the same number. The chapter says so on p. 31 for the tile pattern, and it is worth building the whole of section 9 around.
"y = x is the standard against which all steepness is measured." It is the chapter's chosen reference because it leans equally towards both axes, which is a convenient landmark and not a law. Say why it was chosen.

Questions to check understanding

Identify the slope and the y-intercept of a given equation, including when the equation must first be rearranged into the form y = ax + b — End-of-Chapter starred item 7 on p. 37 sets 2y = 4x + 7, 5y = 6x – 10 and 3y = 6x – 11 precisely so that rearranging is required
Give, as a coordinate pair, where a given line meets the vertical axis
Decide which of a listed set of lines are parallel — asked directly at the end of End-of-Chapter item 7 on p. 37
Draw a family of lines and describe the effect of the constant that varies
Find a linear polynomial whose graph is parallel to another and passes through a given point — End-of-Chapter starred item 13 on p. 39
State what a set of functions of a given form have in common — End-of-Chapter starred item 14 on p. 39 asks this of f(x) = ax + a with a > 0

Examples worth working on the board

Inputs below. All figure descriptions are read off the printed pages. Values marked verified are worked out here; no answers are printed in this chapter and the book has no appended key.

Example 14's family, drawn separately (Fig. 2.8, p. 30). Three stacked panels on printed squared paper, each with axes running –6 to 6 across and –3 to 3 up. The panels carry y = ½x (dark blue), y = x (red) and y = 2x (black), each labelled beside its own line, and no points are labelled. That last remark is the body prose on p. 30 that introduces the figure; the caption itself is only the figure number.
The same three on one set of axes (Fig. 2.9, p. 31). One panel, axes –6 to 6 across and –3 to 3 up, with all three lines drawn and labelled and all three passing through the same point at the centre. The chapter asks what the reader can conclude about y = ax with a > 0 as a varies, and specifically what happens for a > 1 and for a < 1. Its printed hint suggests also plotting y = 3x and y = ⅓x.
The chapter's conclusions about y = ax (p. 31). Every such line passes through the origin (0, 0). When a > 1 the line is steeper than y = x, which leans equally towards both axes. When a < 1 it leans more gently than y = x. And a is named the slope of y = ax, with a pointer forward to a later chapter on linear equations for more about slope.
Why the origin, argued rather than observed. Verified, and an added argument — substituting x = 0 into y = ax gives y = 0 whatever a is, so (0, 0) satisfies every equation of that form and therefore lies on every one of those lines. The chapter reports the fact from the picture; the substitution is one line and settles it for coefficients nobody has drawn.
Numbers for the steepness scene. Verified — take one step right from the origin. y = ½x rises 0.5, y = x rises 1, y = 2x rises 2. So the rise per unit across is the coefficient. That is the cleanest reading of slope available at this level, and it makes "steeper" measurable rather than visual.
Example 15's family, drawn separately (Fig. 2.10, p. 32). Three stacked panels, axes –7 to 7 across and –4 to 4 up. They carry y = –⅓x (dark blue), y = –x (red) and y = –3x (black), each labelled, and again with no points labelled.
The same three together (Fig. 2.11, p. 33). One panel, axes –7 to 7 across and –4 to 4 up, all three lines drawn and labelled, all through the centre. The chapter asks what can be concluded about y = –ax with a > 0 as a varies, and again asks about a > 1 and a < 1.
The chapter leaves the negative case open. I checked the printed pages 32, 33 and the chapter summary on pp. 39–40: the questions set alongside Fig. 2.11 — printed in the paragraph that introduces it, above the figure, as with Fig. 2.9 on p. 31 — are not answered in print. Verified, an added resolution — steepness is governed by the size of a with the sign set aside, so y = –3x is as steep as y = 3x and steeper than y = –x, while y = –⅓x is the shallowest of the three. The sign decides which way the line leans, not how sharply. Present this as the answer being worked out, not as printed text.
Think and Reflect, p. 33. Contrast the graphs of y = 3x + 1 and y = –3x + 1. Hand over both equations. Verified — both cross the vertical axis at (0, 1); one climbs and one falls; both are equally steep. A neat item, because it separates the two things a controls — direction and steepness — in a single pair.
The sign of the slope, tied back to §2.4 (p. 31). The chapter states that linear growth appears as a line of positive slope and linear decay as a line of negative slope.
The slope as the constant gap (p. 31). The chapter states that the tile sequence 1, 3, 5, 7 … has consecutive gaps of 2, that the relationship between stage number and tile count is y = 2x – 1, that its slope is 2, and therefore that the slope represents the fixed gap between consecutive terms. Verified — the sentence carrying this is garbled in print, reading as though "is 2" appears twice. The claim itself is unambiguous and is the most valuable connection in the chapter, joining §2.3's tables to §2.6's drawings.
Example 16's family, drawn separately (Fig. 2.12 A on p. 33, B and C on p. 34). Three panels on squared paper, axes running –6 to 7 across and –2 to 5 up. Fig. 2.12 A carries y = 2x – 1 (dark blue), B carries y = 2x + 1 (red) and C carries y = 2x + 5 (black), each labelled along its line.
The same three together (Fig. 2.13, p. 34). One panel, axes –6 to 7 across and –2 to 5 up, with all three lines drawn and labelled and visibly never meeting. This is the figure that evidences the parallel rule.
The hint that names the experiment (p. 35). It states that in these three equations a = 2 while b takes the values –1, 1 and 5. The Think and Reflect box just above asks what can be concluded about y = ax + b when a is fixed and b varies.
Fig. 2.14 and the intercepts (p. 35). One panel on squared paper, axes –3 to 6 across and –2 to 7 up, carrying y = 2x + 5 (red), y = x + 3 (black) and y = 3x – 2 (dark blue), each labelled, with three points marked on the vertical axis and lettered A, B and C. The chapter states that y = 2x + 5 crosses at A (0, 5), y = x + 3 at B (0, 3), and y = 3x – 2 at C (0, –2).
A warning about Fig. 2.14. I read this off p. 35: its three lines have coefficients 2, 1 and 3 — all different — so they are not parallel. Fig. 2.14 is the evidence for reading b off the axis, and Fig. 2.13 is the evidence for the parallel rule. Anyone who reaches for the more visually striking Fig. 2.14 to illustrate parallel lines will illustrate the opposite.
The intercept rule (p. 35). The chapter states that a line written as y = ax + b crosses the vertical axis at (0, b); that b is called the y-intercept; that the y-intercept of y = x + 3 is 3, meaning the crossing sits three units up the axis from the origin; and that for y = 3x – 2 it is –2, two units down from it.
Why (0, b) rather than anything else. Verified, an added argument — the vertical axis is exactly the set of points whose x-coordinate is 0, so substitute x = 0 into y = ax + b and the ax term vanishes, leaving y = b. The chapter reports the rule from three drawn cases; the substitution proves it for all of them at once, and it is the same one-line move that settled the origin in section 3.
The three printed conclusions (p. 36). After Figs. 2.8 to 2.14 the chapter states: a is the slope and b the y-intercept; changing a with b held fixed changes the slope and leaves the y-intercept alone; and changing b with a held fixed shifts the line while keeping it parallel to the original, so lines of equal slope and different y-intercepts are parallel.
Exercise Set 2.6 (p. 36), one item in five parts. Draw each set and reflect on the roles of a and b: (i) y = 4x, y = 2x, y = x. (ii) y = –6x, y = –3x, y = –x. (iii) y = 5x, y = –5x. (iv) y = 3x – 1, y = 3x, y = 3x + 1. (v) y = –2x – 3, y = –2x, y = 2x + 3.
What each part is doing, and the one that does something else. Verified — (i) varies a positive coefficient with b at zero; (ii) does the same with negative coefficients; (iii) is a mirror pair of equal steepness; (iv) holds the coefficient at 3 and moves b through –1, 0 and 1, so all three are parallel. Part (v), however, lists coefficients –2, –2 and +2: the first two are parallel and the third is the mirror image of them. Every other part in the item is internally consistent, so (v) is either a deliberate odd-one-out or a sign slip.
A caution on the word "distance" (pp. 35–36 and 40). The chapter reaches for length-and-distance wording for b in both places: p. 35 calls b a length and reads the positive case as a crossing a certain distance from the origin, and the summary on p. 40 does the same. What pp. 35–36 add — and the summary drops — is a direction: the sentence completing the negative case, printed at the top of p. 36, gives the crossing as a distance from the origin in the negative direction, where p. 35 alone ends with the bare value –2. So the contrast is not loose wording against exact wording; it is the same wording with and without the direction that rescues it. Verified — a distance cannot be –2, so b is a signed coordinate rather than a distance.

Figures to have open

A coordinate grid on which several lines can be drawn, labelled and shown step by step — one where the coefficient can be varied with the crossing point pinned, and one where the constant can be varied with the direction pinned. This replaces Figs. 2.8 to 2.14 (pp. 30–35) and carries the topic. Redraw as clean schematics rather than reproducing the book's printed green graph paper.
A one-step-right overlay that measures the rise on each line, so steepness is read as a number rather than judged by eye. Standard schematic; the chapter compares steepness only visually.
A parallel-family panel built from y = 2x – 1, y = 2x + 1 and y = 2x + 5 — the content of Fig. 2.13 (p. 34), not Fig. 2.14 (p. 35), whose three lines have three different coefficients.
A separate intercept panel built from y = x + 3, y = 2x + 5 and y = 3x – 2 with the three crossings lettered — the content of Fig. 2.14 (p. 35), used only for reading b.
A side-by-side panel for Exercise Set 2.6 parts (iv) and (v), with the odd line in (v) marked. Standard schematic.

Where this sits in the book

NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.6 "Visualising linear relationships", pp. 29–36. Fig. 2.8 is on p. 30; Fig. 2.9 and the conclusions about y = ax, the slope, the growth-decay sign link and the slope-as-gap statement are all on p. 31; Fig. 2.10 is on p. 32; Fig. 2.11, the Think and Reflect contrast and Fig. 2.12 A are on p. 33; Fig. 2.12 B, Fig. 2.12 C and Fig. 2.13 are on p. 34; the Think and Reflect box, the hint naming a = 2, Fig. 2.14 and the intercept rule are on p. 35; the three numbered conclusions and Exercise Set 2.6 are on p. 36.
Examples 14, 15 and 16 are introduced on pp. 29, 31 and 33 respectively.
Backward links inside the chapter: the tile pattern and its constant gap are on pp. 21–22 (A constant difference is the signature of a linear pattern); linear growth and decay are §2.4, pp. 24–25 (Linear growth and linear decay); plotting from two points is pp. 27–29 (Why two points are enough to draw the line).
Forward pointer out of the chapter: p. 31 names a later chapter on linear equations as the place slope is developed.
End-of-Chapter Exercises, starred items 7, 9, 10, 11, 13 and 14 on pp. 37–39.
Chapter summary, pp. 39–40, restates the slope, the y-intercept, the b = 0 case, the growth-decay sign link and the parallel rule.