Two taxis, two price lines: when does the cheaper one switch?

Gradient and the forms of a line

• Gradient m=x2​−x1​y2​−y1​​m = (y₂ − y₁)/(x₂ − x₁): the rate of change of $y$y per unit of $x$x.
• Gradient-intercept form: $y=mx+c$y = mx + c, where $c$c is the $y$y-intercept.
• Point-gradient form: y−y1​=m(x−x1​)y − y₁ = m(x − x₁) through the point (x1​,y1​)(x₁, y₁).
• General form: $ax+by+d=0$ax + by + d = 0.
• A road climbing 1.2 m over 15 m horizontally has gradient 151.2​=0.081.2/15 = 0.08, an 8% incline.

Parallel and perpendicular lines

• Parallel lines have equal gradients: m1​=m2​m₁ = m₂.
• Perpendicular lines: m1​×m2​=−1m₁ × m₂ = −1, so m2​=−m1​1​m₂ = −1/m₁ (the negative reciprocal).
• To read a gradient from general form, rearrange to $y=mx+c$y = mx + c. For $2x+5y-7=0$2x + 5y − 7 = 0: y=−52​x+57​y = −2/5x + 7/5, so m=−52​m = −2/5.
• Any line perpendicular to it has gradient 25​5/2.

Functions, domain, range and inverses

• A function gives each input exactly one output. Notation: $f(x)$f(x), $v(t)$v(t), $C(n)$C(n).
• The domain is the set of allowed inputs; the range is the set of outputs. f(x)=2−x​f(x) = √(2 − x) has domain $x\le 2$x ≤ 2 and range $f(x)\ge 0$f(x) ≥ 0.
• The inverse function f−1f⁻¹ undoes $f$f: solving $f(x)=10$f(x) = 10 is the same as finding f−1(10)f⁻¹(10).
• The graph of f−1f⁻¹ is the reflection of $f$f in $y=x$y = x. Only one-to-one functions have inverses; the domain of f−1f⁻¹ is the range of $f$f.

Key features of graphs, with technology

• Sketch: the shape, with axes and key features labelled. Draw: accurate, to scale.
• Key features: intercepts, zeros (roots of $f(x)=0$f(x) = 0), maximum and minimum points, vertex, symmetry, vertical and horizontal asymptotes.
• To solve $f(x)=g(x)$f(x) = g(x), graph both on your GDC and find the intersection points. A GDC also graphs sums and differences such as $y=f(x)-g(x)$y = f(x) − g(x), whose zeros are the same $x$x-values.

A phone loses 20% of its value every year. Is it ever worth nothing?

Linear and piecewise linear models

• A linear model $f(x)=mx+c$f(x) = mx + c fits a quantity that changes by the same amount per unit of $x$x: $m$m is the rate, $c$c the start value.
• A piecewise model uses different rules on different parts of the domain, like a phone plan with a data allowance, then a charge per GB.
• Always state a reasonable domain: data used cannot be negative.

Exponential growth and decay models

• An exponential model fits a quantity that changes by the same percentage each step: f(x)=kax+cf(x) = ka^x + c or f(x)=ka−x+cf(x) = ka^(−x) + c (with $a>0$a > 0), or f(x)=kerx+cf(x) = ke^rx + c.
• In kaxka^x: $a>1$a > 1 is growth, $0<a<1$0 < a < 1 is decay. A 20% fall each year means $a=0.8$a = 0.8; a 5% rise means $a=1.05$a = 1.05.
• The starting value is $f(0)=k+c$f(0) = k + c.
• The graph has a horizontal asymptote $y=c$y = c: decay models get ever closer to $c$c without reaching it.

Direct and inverse variation

• Variation models have the form f(x)=axnf(x) = axⁿ, $n\in \mathbb{Z}$n ∈ ℤ.
• $n>0$n > 0: direct variation, such as y=ax2y = ax² ($y$y varies as the square of $x$x).
• $n<0$n < 0: inverse variation, such as y=x2a​=ax−2y = a/x² = ax⁻². The $y$y-axis is a vertical asymptote.
• Find $a$a from one data point: if y=ax−2y = ax⁻² and $y=45$y = 45 when $x=2$x = 2, then a=45×22=180a = 45 × 2² = 180, and $y=20$y = 20 when $x=3$x = 3.
• Choose a model from the context and the shape of the data, and beware of extrapolating far beyond the data.

The tide rises and falls like a wave. When is the harbour too shallow?

Quadratic models

• f(x)=ax2+bx+cf(x) = ax² + bx + c, $a\ne 0$a ≠ 0, fits data that rise then fall (or fall then rise): a thrown ball, an arch, the area of a fenced plot.
• The vertex gives the maximum ($a<0$a < 0) or minimum ($a>0$a > 0); the axis of symmetry is x=−2ab​x = −b/2a.
• The zeros (use your GDC) often mark where the model starts or ends, such as when a ball lands.
• Choose a reasonable domain: a height cannot be negative, so the model stops at the zero.

Cubic models and finding parameters

• f(x)=ax3+bx2+cx+df(x) = ax³ + bx² + cx + d fits data with up to two turning points, such as the volume of an open box cut from a sheet.
• A box from a 30 cm by 20 cm sheet with squares of side $x$x cut out has $V=x(30-2x)(20-2x)$V = x(30 − 2x)(20 − 2x), domain $0<x<10$0 < x < 10. A GDC gives the maximum: $V\approx 1,056$V ≈ 1,056 cm³ at $x\approx 3.92$x ≈ 3.92.
• Finding parameters: substitute known points into the model. Three points give three linear equations in $a$a, $b$b, $c$c: solve them with your GDC.

Sinusoidal models

• Amplitude ∣a∣=2max−min​|a| = (max − min)/2; principal axis y=d=2max+min​y = d = (max + min)/2.
• Period =b360∘​= (360°)/b, with $x$x in degrees: the time for one full cycle.
• $a\cos (bx)+d$acos (bx) + d with $a>0$a > 0 starts at a maximum; with $a<0$a < 0 it starts at a minimum.

Big ideas

• A line has gradient m=x2​−x1​y2​−y1​​m = (y₂ − y₁)/(x₂ − x₁) and can be written $y=mx+c$y = mx + c, y−y1​=m(x−x1​)y − y₁ = m(x − x₁) or $ax+by+d=0$ax + by + d = 0; parallel means m1​=m2​m₁ = m₂, perpendicular means m1​m2​=−1m₁ m₂ = −1.
• A function has a domain (inputs) and a range (outputs); f−1f⁻¹ undoes $f$f, exists for one-to-one functions and is the reflection of $f$f in $y=x$y = x.
• Key features (intercepts, zeros, maximum and minimum, asymptotes) and intersections come from a GDC or from solving $f(x)=g(x)$f(x) = g(x).
• Same amount each step: linear $f(x)=mx+c$f(x) = mx + c (piecewise if the rule changes). Same percentage each step: exponential kax+cka^x + c.
• An exponential model kax+cka^x + c has horizontal asymptote $y=c$y = c; decay approaches it but never reaches it.
• Variation axnaxⁿ: find $a$a from one point. Choose models from context and data shape; beware extrapolation.
• Quadratic models: vertex for the maximum or minimum, zeros for start and end, and a domain that makes sense.
• Find a model's parameters by substituting points and solving the linear equations with your GDC.
• Sinusoidal models: amplitude 2max−min​(max − min)/2, principal axis y=2max+min​y = (max + min)/2, period b360∘​(360°)/b.