Reference Sheet: Programme F.4 — Graphs

Rules, procedures and drill, in the order of the book's programme. Every rule links to the unit in the lesson that derives it — if a rule here looks arbitrary, the link is where it stops being arbitrary.

1. Equations in graphing form

From Unit 01.

An equation in two variables is a conditional equation: true only for some pairs of values. Transpose it to make y the subject, "y= an expression in x". Then x is the independent variable (chosen freely) and y the dependent variable (restricted to the value the expression gives). Making y the subject is a convention, not a necessity.

Examples. x-y=1 gives y=x-1. x+2y=6 gives 2y=6-x, so y=3-\frac x2.

2. Domain restrictions

From Unit 01.

Transposing can restrict x as well as y: a square root needs a non-negative number under it, and it gives two values of y, one of each sign.

Examples.

Notation: the book writes \sqrt x for the positive root and x^{1/2} for both roots. In these notes both mean the non-negative root, and the second value is written with an explicit \pm.

3. Ordered pairs

From Unit 01.

Choose a value of the independent variable, calculate the dependent one, and write them as (x,y) — independent first. The order matters: (2,4) and (4,2) are different points.

Example. y=3x-1 for x=-1,0,1,2: (-1,-4), (0,-1), (1,2), (2,5).

4. Cartesian axes

From Unit 01.

Two perpendicular numbered lines crossing at their common zero: the Cartesian coordinate frame. The horizontal axis is always the independent-variable axis (x-axis), the vertical the dependent-variable axis (y-axis). The two scales need not be equal; choose them to fill the paper.

5. Drawing a graph

From Unit 01.

Plot the ordered pairs as isolated points. Infinitely many choices of x would give points merging into a continuous line, the graph of the equation; in practice, plot enough points and join them with a smooth curve, not a ruler. The straight-line join between points \delta x apart is in error by an amount that shrinks like (\delta x)^2.

Examples. y=x^2 for x=-3,\ldots,3: 9,4,1,0,1,4,9 — a parabola. y=x+1 for x=-4,\ldots,4: -3,-2,\ldots,5 — a straight line.

6. Tables built term by term

From Unit 01.

Give each term of the expression its own row (or column), evaluate it at every x, then add. It works for any equation, polynomial or not.

Example. y=x^2+\frac1x:

7. Asymptotes

From Unit 02.

When a curve approaches a second curve or line arbitrarily closely without meeting it, the second is an asymptote to the first. A vertical asymptote x=a appears where a denominator is zero: the values grow without bound on each side. Not all asymptotes are vertical, or straight.

Beyond the book: a horizontal asymptote y=L is a level the values settle towards as x grows large, and a curve may cross one (y=\frac{x}{x^2+1} crosses y=0 at the origin).

Examples. y=\frac1{1-x}: y(0.9)=10, y(1.1)=-10; vertical asymptote x=1, horizontal asymptote y=0. y=\frac3{x+2}: y(-1.9)=30, y(-2.1)=-30; vertical asymptote x=-2. y=x^2+\frac1x approaches the parabola y=x^2 far from the origin.

8. Piecewise equations

From Unit 02.

Different expressions apply on different parts of the x-axis; which one to use depends on where x lies.

Example. y=x^2 for -5\le x<0 and y=x for x\ge0: a parabola arm joined at the origin to a straight line. Both pieces give 0 at x=0, so the graph is unbroken.

9. Discontinuities

From Unit 02.

A gap in a graph is a discontinuity. Take care when joining points: never join across a place where the formula has no value or where a piecewise formula changes. A dashed line across a gap guides the eye but is not part of the graph. Mark endpoints with a closed dot (belongs to the graph) or an open circle (does not).

Example. y=1 for x\le2, y=-1 for x>2: two horizontal lines; closed dot at (2,1), open circle at (2,-1).

Beyond the book: an infinite discontinuity is an asymptote; a jump is a step like this one; a hole is a single missing point, as in y=\frac{x^2-1}{x-1}, which is the line y=x+1 without the point (1,2).

10. Spreadsheets: rows, columns and the active cell

From Unit 03.

A worksheet is a grid of cells. Columns are lettered from A, rows numbered from 1, and a cell's address is its column letter then its row number: P123 is column P (the 16th letter), row 123. The active cell is outlined by the cursor; move it with the arrow keys or the mouse.

11. Text, numbers and formulas

From Unit 03.

Type text or a number into the active cell and press Enter. A formula starts with =; * is multiplication and ^ a power. A formula recalculates whenever a cell it uses changes.

Examples. 12 in C15 and =3*C15 in C16 shows 36; change C15 to 5 and C16 shows 15. y=2x^2-3 with x in A1: =2*A1^2-3.

12. Clearing entries

From Unit 03.

Select a cell and press backspace. For a block, highlight it by dragging and cut it (or Edit → Clear → All).

13. Constructing a Cartesian graph

From Unit 03.

1. Type the first x-value in A1; highlight the column down to the last row; Fill → Series → Columns, with the step value.
2. Type the formula in B1, using A1 for x.
3. Copy it down by dragging the small square at the bottom right of B1. The relative address A1 becomes A2, A3, … in the copies.
4. Highlight both columns; Insert → X-Y Scatter → "Scatter with Smooth Lines".

Example. y=(x-2)^3, -1\le x\le5, step 0.3: 21 rows, A21 =5. B1 =(A1-2)^3 shows -27; B2 shows (-2.7)^3=-19.683; B11 (x=2) shows 0.

Beyond the book: use X-Y Scatter, not a Line chart, which spaces the rows evenly whatever their x-values.

14. Displays and new equations

From Unit 03.

Chart Design → Quick Layout; delete the "Series 1" legend; type the axis titles; resize by dragging the handles. To plot a new equation, overtype B1 and copy it down again: the chart redraws itself.

15. Reading roots from a plot

From Unit 03 and Unit 01; the factor theorem is F.3 Unit 04.

Where the graph meets the x-axis, y=0: those x-values are the roots. A plot suggests them; the factor theorem confirms them.

16. Stray lines across breaks

From Unit 03 and Unit 02.

"Smooth lines" are drawn straight across asymptotes and jumps. Insert an empty row (Insert → Rows) between the two rows that straddle the break, and the chart leaves a gap there.

Examples. y=\frac1{1-x} on the -1 to 5, step-0.3 grid: the break at x=1 falls between 0.8 and 1.1, so insert at A8. The step 1/-1 at x=2: insert at A12, between 2 and 2.3. The magnifier \frac{10}{10-u} for u=1,3,\ldots,21: insert between u=9 and u=11; with an 8 cm lens, between u=7 and u=9.

17. Less than or greater than

From Unit 04.

a<b: a is less than b, to its left on the number line. 3<5, -2>-4. An inequality in x and y such as y>x is satisfied, for each x, by infinitely many y: its graph is a region, not a line.

18. Regions above and below a curve

From Unit 04.

The curve y=f(x) separates the plane into two regions:

It works because every vertical line meets y=f(x) exactly once: compare y with f(x) at the same x.

Examples. y>x^2: above the parabola; (1,3) is in it, since 3>1. y<x^3-2x^2: below the cubic; (2,-1) is in it, since f(2)=0 and -1<0.

19. Making y the subject first

From Unit 04; the sign switch is proved in Unit 06.

Rearrange to y>\ldots, y<\ldots before reading off the region. Adding, subtracting, and multiplying or dividing by a positive number keep the direction; multiplying or dividing by a negative number reverses it.

Examples. 2x+3y>6: 3y>6-2x, y>2-\frac23x — above the line. 4x-2y\le8: -2y\le8-4x, divide by -2 and switch: y\ge2x-4 — on or above the line.

20. Beyond the book: boundaries and test points

From Unit 04.

Draw the boundary dashed for < or > (excluded) and solid for \le or \ge (included). Confirm the side with a test point not on the boundary: if the inequality holds there, that side is the region. A test point also handles regions that are not above or below one curve.

Examples. y\le x^2-3x+2: solid boundary; (0,0) gives 0\le2, in; (1.5,0) gives 0\le-0.25, out. x^2+y^2\le25: (0,0) gives 0\le25, so the region is the inside of the circle of radius 5.

21. Absolute value or modulus

From Unit 05.

A number's distance from zero on the number line is its absolute value or modulus, |x|, read "mod x":

Examples. |-5|=5, |3|=3, |-2.41|=2.41, |13.6|=13.6, |-4|=-(-4)=4.

22. The modulus of an expression

From Unit 05.

Split where the expression inside is zero: keep it where it is non-negative, negate it where it is negative.

Examples.

23. Graphs of |x| and |x-a|

From Unit 05.

y=|x| is two straight lines forming a V with its point at the origin: the part of y=x below the axis reflected upwards. y=|x-a| is the same V with its point at x=a.

Example. y=|x-2|: point at (2,0); y=2 at x=0 and at x=4.

24. Less-than inequalities

From Unit 06.

For a>0: (a) |x|<a\Rightarrow-a<x<a; (b) |x\pm b|<a\Rightarrow-a\mp b<x<a\mp b. Graphically, the part of the V below the line y=a. Write the double inequality, then work on all three parts at once.

Examples.

25. Operating on inequalities: switching the sign

From Unit 06.

Add or subtract across all parts freely; multiply or divide by a positive number freely. Multiplying or dividing by a negative number reverses the order of the number line, so switch every sign: (e) -ax>b\Rightarrow x<-\frac ba; (f) -ax<b\Rightarrow x>-\frac ba.

Examples.

26. Greater-than inequalities

From Unit 06.

For a>0: (c) |x|>a\Rightarrow x>a or x<-a; (d) |x\pm b|>a\Rightarrow x>a\mp b or x<-a\mp b. Graphically, the two arms of the V above the line y=a. Solve the two pieces separately; never combine them into one double inequality.

Examples. |x|>4: x>4 or x<-4. |x+3|>7: x+3>7 or x+3<-7, so x>4 or x<-10. |5-x|>2: -x>-3 or -x<-7, so x<3 or x>7. |7-2x|>9: -2x>2 or -2x<-16, so x<-1 or x>8.

27. Interaction: relative and absolute addresses

From Unit 03 and Unit 05.

A1 is a relative address, "the cell in this position relative to me", and changes when the formula is copied. $A$1 is an absolute address, that actual cell, and never changes. Mixing the two makes a sheet interactive: constants in absolute cells, variables in relative ones.

Example. B1: =ABS($C$1*(A1+$D$1))+$E$1 plots y=|a(x+b)|+c with a in C1, b in D1, c in E1. Start from 1,0,0 (the graph of |x|). b=2 moves the V 2 units left; c=3 moves it 3 units up; a=4 steepens the arms to gradients \pm4. The point of the V is at (-b,c).

28. Beyond the book: modulus as distance

From Unit 05 and Unit 06.

|x-c| is the distance between x and c. So |x-c|<r means "within r of c", c-r<x<c+r, and |x-c|>r means "further than r from c". A tolerance |L-L_0|\le\Delta L reads directly as L_0-\Delta L\le L\le L_0+\Delta L. And |u|=|-u|, so |7-2x|=|2x-7|: rewriting with a positive coefficient avoids the sign switch.

Examples. |L-25|\le0.05 mm: 24.95\le L\le25.05. |700-20t|<300 is 20|t-35|<300, |t-35|<15: 20<t<50.

Traps

• Joining across an asymptote. The table of \frac1{1-x} joined by a ruler crosses the axis at x=1, where the function has no value at all. Find where the formula has no value before joining.
• The dashed line is part of the graph. At the step y=1 (x\le2), y=-1 (x>2) the function never takes the value 0; a line drawn across the gap is only a guide.
• Only one branch. x^2+y^2=1 gives y=\pm\sqrt{1-x^2}: two values for each x, not one. And forgetting the restriction -1\le x\le1 produces square roots of negative numbers.
• Comparing at the wrong x. "4.8<5, so the lorry fits under the 5 m arch" compares the lorry's corner with the arch at its centre. Compare y with f(x) at the point's own x.
• Reading a region before making y the subject. 4x-2y\le8 is on or above y=2x-4, not below: dividing by -2 switches the sign.
• Not switching the sign. |7-2x|<9 gives -16<-2x<2; dividing by -2 without switching gives 8<x<-1, which no number satisfies, although x=0 gives |7|<9.
• One double inequality for a greater-than. |x|>4 is x>4 or x<-4; writing -4>x>4 describes no number at all.
• A negative distance. |x-3| is not x-3 for every x: at x=1 it is 2, not -2. Split at the zero of the inside.
• Splitting the modulus of a sum. |a+b| is not |a|+|b|: |-3+5|=2, but |-3|+|5|=8.
• A relative address where an absolute one is meant. =(A1-E1)^2 copied down becomes =(A2-E2)^2, and the empty E2 counts as 0. Pin the constant: =(A1-$E$1)^2.

Self-check

1. Make y the subject of x^2+y^2=100, state the values of x allowed, and give the ordered pairs for x=0, \pm6, \pm8, \pm10.
2. Tabulate y=x^3-3x term by term for x=-2,-1,0,1,2, and say from the table where the graph meets the x-axis.
3. For y=\frac{2}{x-3}: find y at 2.9, 3.1, 2.5 and 3.5. What does a ruler line through the points at 2.5 and 3.5 wrongly suggest, and what are the asymptotes?
4. Does y=x+2 (x<1), y=5-x (x\ge1) have a discontinuity? What about y=x+2 (x<1), y=4-x (x\ge1)?
5. Column A holds -2,-1.75,\ldots in steps of 0.25 from A1. Write the formula for B1 to plot y=2x^2-3. What does B7 contain after copying down, and what value does it show?
6. To plot y=(x-h)^2 with h in E1, what goes in B1? What goes wrong if the dollar signs are left out?
7. Describe the region y>x^2-4, and decide whether (1,-2) and (3,4) lie in it.
8. Rearrange 3x-2y\ge6 to read off its region, and say whether the origin lies in it.
9. Write |3x+6| without the modulus sign, and give the point of the V of y=|3x+6|.
10. How does y=|2(x-1)|+3 differ from y=|x|? Find its point and its value at x=-1.
11. Solve |3x+4|<5.
12. Solve |4-3x|\ge10.

Fully worked solutions

1. y^2=100-x^2, so y=\pm\sqrt{100-x^2}, and 100-x^2\ge0 requires -10\le x\le10. Pairs: (0,\pm10), (\pm6,\pm8), (\pm8,\pm6), (\pm10,0) — each \pm in x and in y taken independently, so x=6 gives (6,8) and (6,-8).

2.

The graph meets the axis at x=0, and y changes sign between -2 and -1 and between 1 and 2. (Exactly: x^3-3x=x(x^2-3), so x=0, \pm\sqrt3=\pm1.732.)

3. y(2.9)=\frac2{-0.1}=-20, y(3.1)=20, y(2.5)=-4, y(3.5)=4. The ruler line from (2.5,-4) to (3.5,4) passes through (3,0), suggesting y=0 at x=3; in fact y has no value there. Vertical asymptote x=3; horizontal asymptote y=0.

4. First: as x approaches 1 from the left, x+2 approaches 3; at x=1, 5-1=4. The graph jumps from 3 to 4: a discontinuity, with a closed dot at (1,4) and an open circle at (1,3). Second: x+2\to3 and 4-1=3, so the pieces meet at (1,3): no discontinuity, just a corner.

5. B1: =2*A1^2-3. After copying, B7 contains =2*A7^2-3. A7 =-2+6\times0.25=-0.5, so B7 shows 2\times0.25-3=-2.5.

6. =(A1-$E$1)^2. Without the dollar signs, =(A1-E1)^2 copied down becomes =(A2-E2)^2, =(A3-E3)^2, …; those cells are empty and count as 0, so every row after the first plots y=x^2 instead.

7. The region above the parabola y=x^2-4, which meets the x-axis at \pm2; the boundary is excluded (dashed). At x=1 the curve is at -3, and -2>-3: (1,-2) is in. At x=3 the curve is at 5, and 4>5 is false: (3,4) is not.

8. -2y\ge6-3x; divide by -2 and switch: y\le\frac32x-3 — on or below the line of gradient \frac32 through (0,-3). The origin: 0\le-3 is false, so it is not in the region (check in the original: 0\ge6 is false too).

9. 3x+6\ge0 when x\ge-2: |3x+6|=3x+6 for x\ge-2, and -3x-6 for x<-2. The point of the V is (-2,0), with arms of gradient \pm3.

10. In |a(x+b)|+c: a=2 steepens the arms to \pm2; b=-1 moves the V 1 to the right; c=3 lifts it 3. The point is (1,3). At x=-1: |2(-2)|+3=4+3=7.

11. -5<3x+4<5; subtract 4: -9<3x<1; divide by 3: -3<x<\frac13.

12. 4-3x\ge10 or 4-3x\le-10. First: -3x\ge6, divide by -3 and switch: x\le-2. Second: -3x\le-14, so x\ge\frac{14}3. Answer: x\le-2 or x\ge\frac{14}3=4.667.