Q: What does IP Maths Notes (Lower Sec, Year 1-2): 04) Coordinate Geometry & Graphs cover? A: Understand gradients, intercepts, distance and midpoint formulas, and sketch linear graphs with contextual interpretations.
Coordinate geometry links algebra to visuals. This note refreshes gradient logic, distance and midpoint formulas, equation forms, and graph interpretation.
Learning targets
Compute gradients and intercepts from points or equations.
Convert between point-slope, slope-intercept, and general linear forms.
Apply the distance and midpoint formulas accurately.
Interpret real-world scenarios from line graphs (rate, supply-demand, temperature).
1. Gradient and intercepts
Gradient (slope) between points (x1,y1) and (x2,y2)
:
m=x2−x1y2−y1.
Slope-intercept form: y=mx+c, where c is the y-intercept.
Point-slope form: y−y1=m(x−x1).
Worked example - Gradient & intercept
Find the gradient and y-intercept of the line passing through (2,−3) and (5,6).
m=5−26−(−3)=39=3.
Use point-slope form with (2,−3):
y+3=3(x−2)⟹y=3x−9.
Y-intercept is −9.
2. Distance and midpoint formulas
Distance between two points: d=(x2−x1)2+(y2−y1)2.
Midpoint: M(2x1+x2,2y1+y2)
Example: Points A(−2,5) and B(4,−1).
d=(4−(−2))2+(−1−5)2=62+(−6)2=72=62.
Midpoint:
M(2−2+4,25+(−1))=M(1,2).
3. Parallel and perpendicular lines
Parallel lines share the same gradient.
Perpendicular gradients multiply to −1: if m1×m2=−1, the lines are perpendicular.
Example: The line through (3,−4) perpendicular to y=−21x+1 has gradient 2 and equation y+4=2(x−3)⟹y=2x−10.
4. Graph interpretation
Always label axes with units.
Identify intercepts, turning points, and intersection points.
Connect slope to rate e.g.,distance−timegraphs:slope=speed.
Worked example - Interpreting a distance-time graph
A student jogs 3 km in 18 minutes, rests for 6 minutes, then jogs another 2 km in 10 minutes.
Segment 1 gradient: 0.33=10 km h−1.
Segment 2 gradient: 0 (rest).
Segment 3 gradient: 60102=12 km h−1.
Explain what each segment means in sentences - examiners award communication marks.
5. System of equations via intersection
Graphing y=2x+1 and y=−x+7 reveals intersection at (2,5). Solving algebraically verifies the result.
Practice Quiz
Consolidate gradient calculations, distance–midpoint work, and graph interpretation with the interactive quiz.
Try it yourself
Find the equation of the line through (4,−5) with gradient −43.
Determine the midpoint of the segment joining (7,2) and (−1,−6).
Show that the line joining (2,3) and (6,7) is parallel to the line 2x−2y=1.
Two lines intersect at (1,−2). One has equation y=4x−6. Find the equation of the perpendicular line through the same point.
