Course

Cambridge O Level Mathematics: Graphs and Coordinate Geometry (4024)

Name: Cambridge O Level Mathematics: Graphs and Coordinate Geometry (4024)
Price: 1299 BDT

Straight-line graphs and coordinate geometry, travel and speed-time graphs, graphs of functions, sketching curves and functions, taught step by step for Cambridge O Level Mathematics.

73 lessons13h 20mEnglish

What you'll learn

Plot points and draw any straight line from its equation, including ax + by = c, x = k and y = k
Find the gradient of a line from a graph or from two points, and write the equation of the line
Calculate the length and midpoint of a line segment, and find lines parallel or perpendicular to a given line
Read travel and conversion graphs, find speed and acceleration as gradients and distance as an area
Draw quadratic, cubic, reciprocal and exponential graphs and use them to solve equations
Estimate the gradient of a curve by drawing a tangent, and say what it means in a real situation
Sketch a curve showing its roots, turning point, line of symmetry and asymptotes
Use function notation, and find composite and inverse functions

Requirements

Algebra at O Level standard (expanding, factorising, rearranging and solving equations), or our Algebra course
Pythagoras' theorem, used in the length of a line segment

Course content

73 lessons · 13h 20m

3.1-3.3, 3.5 · Straight-line graphs

Plot points in all four quadrants, draw a straight line from its equation, find its gradient from a graph or two points, and write the equation of a line in the forms y = mx + c, ax + by = c and x = k.

0/13

Cartesian coordinates

A point is (across, up) from the origin, and negative numbers reach the other three quadrants.

Preview10m 2.

Drawing straight-line graphs from a table of values

Choose three x values, work out y for each, plot the points and join them with one ruled line.

Preview12m

Drawing lines of the form ax + by = c

Put x = 0 to find where the line crosses the y-axis, and y = 0 for the x-axis, then join the two points.

12m

Horizontal and vertical lines

Every point on a vertical line has the same x, so its equation is x = k. A horizontal line is y = k.

10m

The gradient of a straight line

Gradient is rise divided by run: positive lines go up to the right, negative lines go down, flat lines have gradient 0.

12m

Gradient from the coordinates of two points

Subtract the y values, subtract the x values in the same order, and divide. No graph is needed.

12m

The equation y = mx + c

m is the gradient and c is where the line crosses the y-axis. Two numbers fix one line.

12m

Finding the gradient and y-intercept from an equation

Rearrange to y = mx + c first. For 5x + 4y = 8 the gradient is -5/4, not 5.

12m

Finding the equation of a line from its graph

Read c where the line crosses the y-axis, measure m with a large triangle, then write y = mx + c.

12m

10.

Finding the equation of a line through two points

Find m from the two points, put one point into y = mx + c to find c, and give the equation fully simplified.

12m

11.

Straight-line graphs: practice

A timed drill of mixed questions: plot, read a gradient, name the intercept and write the equation.

12m

12.

Straight-line graphs: exam practice

Past-paper style questions on drawing and describing straight lines, with full working and mark schemes.

15m

13.

Straight-line graphs: summary

Gradient, intercept and the three forms of a line on one page, with the common mistakes.

3.4, 3.6-3.7 · Length, midpoint, parallel and perpendicular lines

Calculate the length of a line segment with Pythagoras, find its midpoint, and find the gradient and equation of a line parallel or perpendicular to a given line, up to the perpendicular bisector.

0/11

The length of a line segment

Two points make a right-angled triangle, so Pythagoras gives the length between them.

Preview12m 2.

The midpoint of a line segment

Add the two x-coordinates and halve, then add the two y-coordinates and halve.

Preview10m

Using the midpoint to find an end point

The step from one end to the middle is the same as the step from the middle to the other end.

10m

Parallel lines

Parallel lines have the same gradient. Rearrange each equation to y = mx + c before you compare.

10m

The equation of a line parallel to a given line

Keep the gradient, then use the given point to find the new c. Parallel to y = 4x - 1 through (1, -3) is y = 4x - 7.

12m

Perpendicular lines

Turn a line through a right angle and its gradient m becomes -1/m. The two gradients multiply to -1.

12m

The equation of a line perpendicular to a given line

Flip the gradient and change its sign, then use the given point to find c.

12m

The perpendicular bisector of a line segment

It passes through the midpoint at right angles: find the midpoint, flip the gradient, then find c.

14m

Length, midpoint, parallel and perpendicular lines: practice

A timed drill of mixed questions: lengths, midpoints, and parallel or perpendicular gradients and equations.

12m

10.

Length, midpoint, parallel and perpendicular lines: exam practice

Past-paper style questions ending in a perpendicular bisector, with full working and mark schemes.

15m

11.

Length, midpoint, parallel and perpendicular lines: summary

The length and midpoint formulas and the parallel and perpendicular gradient rules on one page.

2.9 · Graphs in practical situations

Read and draw conversion and travel graphs, find speed and acceleration as gradients, find distance as the area under a speed-time graph, and estimate a rate from a tangent.

0/11

Conversion graphs

Go up from one scale to the line, then across to the other scale. The line need not pass through (0, 0).

Preview10m 2.

Travel graphs

On a distance-time graph a flat part means stopped, a line going up means moving away and a line going down means coming back.

Preview12m

Drawing graphs from given data

Pick a scale that uses most of the grid, label both axes with units, and plot each pair from the table.

12m

Speed from a distance-time graph

The gradient of a distance-time graph is the speed. Average speed is total distance divided by total time.

12m

Speed-time graphs

Now the height of the graph is the speed. A flat part means constant speed, not stopped.

10m

Acceleration and deceleration

The gradient of a speed-time graph is the acceleration in m/s². A line going down is a deceleration.

12m

Distance travelled: the area under a speed-time graph

Split the area into triangles, rectangles and trapezia and add them. The total is the distance travelled.

14m

Estimating a rate of change from a tangent

On a curved graph, draw the tangent at the point, pick two far-apart points on it and find its gradient.

14m

Graphs in practical situations: practice

A timed drill of mixed questions: read a conversion, find a speed, an acceleration and a distance.

12m

10.

Graphs in practical situations: exam practice

Past-paper style travel and speed-time questions, with full working and mark schemes.

15m

11.

Graphs in practical situations: summary

What gradient and area mean on each kind of graph, on one page, with the common mistakes.

2.10 · Graphs of functions

Complete a table of values and draw quadratic, cubic, reciprocal, root and exponential graphs, solve equations from a graph, model growth and decay, and estimate the gradient of a curve with a tangent.

0/12

Drawing quadratic graphs

Complete the table, plot every point and join them with one smooth curve, never with straight pieces.

Preview12m 2.

Drawing cubic graphs

An x³ term makes an S-shaped curve. Take care with negative x: (-2)³ is -8.

Preview12m

Drawing reciprocal graphs

y = a/x and y = a/x² have no value at x = 0, so the graph comes in two separate pieces.

12m

Graphs of sums of powers of x

For y = 2x + 3/x² or a curve with √x, work out each term in its own row of the table, then add.

14m

Drawing exponential graphs

In y = ab^x + c, x is the power. The curve flattens towards y = c on one side and grows fast on the other.

12m

Solving equations graphically: roots

The roots of f(x) = 0 are the x-values where the curve crosses the x-axis. Read them to 1 decimal place.

12m

Solving equations graphically: a line and a curve

Rearrange the equation so one side is the curve you drew. The other side is the line to draw.

14m

Exponential growth and decay

Growth multiplies by the same number each step. Decay multiplies by a number less than 1 and never reaches zero.

14m

Estimating gradients of curves by drawing tangents

Draw the tangent at the point with a ruler, make a large triangle on it and divide rise by run.

12m

10.

Graphs of functions: practice

A timed drill of mixed questions: complete a table, name a curve, read a root and estimate a gradient.

12m

11.

Graphs of functions: exam practice

Past-paper style table, graph, line-and-curve and tangent questions, with full working and mark schemes.

15m

12.

Graphs of functions: summary

Each curve shape, the graphical method for equations and the tangent method on one page.

2.11 · Sketching curves

Sketch linear, quadratic, cubic, reciprocal and exponential graphs without a table, showing roots, intercepts, turning points, lines of symmetry and asymptotes, and recognise each graph from its equation.

0/10

Sketching linear graphs

A sketch shows the shape and the key points, not every point. For a line, mark where it crosses both axes.

Preview10m 2.

Sketching quadratic graphs

Factorise to find the roots, put x = 0 for the y-intercept, and the line of symmetry is halfway between the roots.

Preview12m

Turning points by completing the square

Write y = (x - p)² + q. The turning point is (p, q) and the line of symmetry is x = p.

14m

Sketching cubic graphs

y = ax³ + b has one root. y = ax³ + bx² + cx factorises with x outside the bracket, so it passes through (0, 0).

12m

Sketching reciprocal graphs and their asymptotes

y = a + b/x gets close to the lines x = 0 and y = a but never touches them. These lines are asymptotes.

12m

Sketching exponential graphs

y = ar^x + b has one horizontal asymptote, y = b, and crosses the y-axis at a + b.

12m

Recognising graphs from their equations

The highest power, a letter under a fraction or x in the power tells you the family before you plot anything.

12m

Sketching curves: practice

A timed drill of mixed questions: name the curve, find its key points and its asymptotes.

12m

Sketching curves: exam practice

Past-paper style sketch and turning-point questions, with full working and mark schemes.

15m

10.

Sketching curves: summary

The five curve families, their key points and asymptotes, on one page.

2.12 · Functions

Use function notation, state a domain and range, work out f(3) and f(2a), form composite functions such as fg(x), and find an inverse function f⁻¹(x).

0/10

Functions and function notation

A function gives exactly one output for each input. f(x) = 3x - 5 is the rule, and f(4) = 7 is one output.

Preview12m 2.

Domain and range

The domain is the set of inputs allowed. The range is the set of outputs you can get.

Preview12m

Evaluating functions and solving f(x) = k

Replace every x with the input, in brackets. h(-2) and h(2a) use the same rule.

12m

Composite functions

gf(x) means do f first, then g. Work from the inside out: gf(2) = g(f(2)).

12m

Finding a composite function algebraically

Put the whole of f(x), in brackets, in place of x in g. Simplify fully, and fg(x) is usually not gf(x).

14m

Inverse functions

f⁻¹ undoes f: it takes each output back to its input. Only a one-to-one function can be undone.

12m

Finding an inverse function

Write y = f(x), make x the subject, then swap x and y. Check that f(f⁻¹(x)) gives x.

14m

Functions: practice

A timed drill of mixed questions: evaluate, compose, invert and solve.

12m

Functions: exam practice

Past-paper style function questions, with full working and mark schemes.

15m

10.

Functions: summary

Notation, domain and range, composite and inverse functions on one page, with the common mistakes.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.