Mission 1/24Build0/24 ✓
One right, two up
Set the slope to . Watch the slope triangle: when you move 1 unit right, how far does the line rise?
Try it in the simulation — the mission completes on its own when you reach the goal.
Slope
1
m = Δy / Δx ↗
y-intercept
1
b · (0, 1)
x-intercept
−1
(−1, 0)
Inclination
45°
α · m = tan α
Equation
y=x+1
y=mx+b · m = 1, b = 1
Table of values
| x | y | Δy |
|---|---|---|
| −2 | −1 | |
| −1 | 0 | +1 |
| 0 | 1 | +1 |
| 1 | 2 | +1 |
| 2 | 3 | +1 |
Constant difference: Δy = m = 1
Setting up the scene…
Concepts
Slope: rate of change
Slope tells you how much y changes when x grows by one. You can compute it from any two points on the line, and it is the same everywhere — the slope triangle rides along with the tracer but its ratio never changes.
m=ΔxΔy=x2−x1y2−y1
Slope–intercept form
y=mx+b m sets the steepness and direction; b is the height where the line crosses the y-axis (the y-intercept). Changing b never rotates the line — it slides it up or down, parallel to itself.
Angle of inclination
If α is the angle the line makes with the positive x-direction, then m=tanα. m=1⇒45∘, m=0⇒ flat, m<0⇒ an obtuse angle. Angle and slope are not proportional: m=2 gives α≈63.4∘.
m=tanα,0∘≤α<180∘
Where a line crosses the axes
On the y-axis x=0, so the crossing is (0,b). On the x-axis y=0: 0=mx+b⇒x=−mb. If m=0 and b=0 the line is parallel to the x-axis and never meets it.
The pattern in a table of values
When x grows one step at a time in a table of values, a line's differences (Δy) always equal the slope m. For a parabola the first differences change, but the second differences (Δ2y) are constant and equal 2a. On a wide screen this table updates live at the top right.
Δ2y=2a
Two ways to write a parabola
Standard form y=ax2+bx+c shows the y-intercept (c) at a glance. Vertex form y=a(x−h)2+k shows the vertex (h,k). If a>0 the arms open up, if a<0 down; the larger ∣a∣, the narrower the parabola. h=−2ab,k=f(h)=c−4ab2
Axis of symmetry
A parabola is a mirror image of itself across the vertical line x=h through its vertex — the glowing plane on the board is that mirror. Watch the tracer's twin: points equally far from the axis have equal y values. f(h−d)=f(h+d)
The discriminant: how many roots?
The roots of ax2+bx+c=0 are where the parabola crosses the x-axis. Δ>0: two roots. Δ=0: one (double) root — the parabola just touches the axis. Δ<0: no real roots — the parabola still exists, it just misses the axis.
Δ=b2−4ac,x1,2=2a−b±Δ
Inside vs. outside
Numbers outside (a, k) act on y: k slides up or down, a stretches vertically. Numbers inside (b, h) act on x and work backwards: (x−3) moves the graph right, b=2 squeezes it horizontally.
g(x)=af(b(x−h))+k
Reflections and key points
a<0: the graph flips over the x-axis (it does not slide down!). b<0: it flips over the y-axis. Every key point moves by one rule — the dashed arrows on the board show each journey.
(x,y)↦(h+bx,ay+k)
The unit circle
On a circle of radius 1, the point at angle θ has coordinates P(cosθ,sinθ). The blue horizontal side is cosθ, the orange vertical side is sinθ. By Pythagoras, always:
cos2θ+sin2θ=1
Radians: arc length
An angle of θ radians cuts an arc of length exactly θ on the unit circle. The circumference is 2π, so 360∘=2π and 180∘=π. The green arc is laid out along the wave's axis with the same length.
θrad=θ∘⋅180∘π
Signs by quadrant
Quadrant I: cos and sin both positive; II: cos<0<sin; III: both negative; IV: sin<0<cos. The size comes from the reference angle to the x-axis: sin150∘=sin30∘.
sin(180∘−θ)=sinθ,cos(180∘−θ)=−cosθ
From circle to wave
Plot the point's height (sinθ) against the angle laid out on a horizontal axis and a sine wave appears. When the point completes a turn the wave repeats: the period is 2π. The cosine wave has the same shape, just 2π ahead.
The four knobs of a wave
∣A∣ is the amplitude (midline to crest), T=∣B∣2π the period, C the phase shift (right if positive), y=D the midline. The highest value is D+∣A∣, the lowest D−∣A∣.
y=Asin(B(x−C))+D
The phase-shift trap
The wave sin(2x−2π) is shifted 4π right, not 2π. Factor out B before reading the shift:
sin(Bx−φ)=sin(B(x−Bφ))
The helix: a wave in 3D
A point spinning on a circle while moving steadily along x traces a helix. Looking down the axis you see a circle; from the side, a sine wave — the wave on the board is the helix's shadow. Springs and screw threads have this shape. Try the Helix view.
(x, Asinφ+D, Acosφ),φ=B(x−C)
Skills and curriculum
Slope and rate of change
Grades 8–10Read slope as "how much y changes when x grows by one" and compute it from two points.
Not started
MEB Grade 8 · Linear Equations · M.8.2.2.6MEB Grade 10 · Analytic Examination · MAT.10.6.2CCSS 8.EE.B.5CCSS 8.F.B.4
Linear functions
Grades 8–9Graph y = mx + b and explain how slope and intercept shape the line.
Not started
MEB Grade 8 · Linear Equations · M.8.2.2.4, M.8.2.2.5, M.8.2.2.6MEB Grade 9 · Quantities and Changes · MAT.9.2.1CCSS 8.F.A.3CCSS HSF-IF.C.7a
Features of a parabola
Grades 9–10Find the vertex, axis of symmetry and direction of opening of y = ax² + bx + c.
Not started
MEB Grade 10 · Quantities and Changes · MAT.10.4.2CCSS HSF-IF.C.7aCCSS HSF-IF.C.8a
Roots and the discriminant
Grades 10–11Use the sign of Δ = b² − 4ac to tell how many roots exist and whether the graph crosses the x-axis.
Not started
MEB Grade 10 · Quantities and Changes · MAT.10.4.2, MAT.10.4.6CCSS HSA-REI.B.4bCCSS HSA-SSE.B.3a
Graph transformations
Grades 10–11Explain how a·f(x − h) + k shifts, stretches and reflects a graph.
Not started
MEB Grade 10 · Quantities and Changes · MAT.10.4.2, MAT.10.4.3, MAT.10.4.4CCSS HSF-BF.B.3
Unit circle, sine and cosine
Grades 10–11Read a unit-circle point's coordinates as cos θ and sin θ, and unwrap them into a wave.
Not started
MEB Grade 10 · Geometric Shapes · MAT.10.1.1MEB Grade 11 · Quantities and Changes (1) · MAT.11.3.1CCSS HSF-TF.A.2
Amplitude, period and phase
Grades 11–12Identify amplitude, period, phase shift and midline in y = A·sin(B(x − C)) + D.
Not started
MEB Grade 11 · Quantities and Changes (1) · MAT.11.3.1CCSS HSF-IF.C.7eCCSS HSF-TF.B.5