0580

Further Graphs and Tangents

Coordinate Geometry and Graphs

The families you must know

  • Being able to match an equation to the right picture saves plotting and catches plotting errors
  • Linear, written y = mx + c, or equivalently as ax + by = c, is a straight line
  • Quadratic, written y = ax² + bx + c, is a single u-shape or n-shape with one turning point
  • Reciprocal, written y = a ÷ x or y = a ÷ x + b, is two separate branches that never meet the axes
  • Cubic, written y = ax³ + b or y = ax³ + bx² + cx, has a long S-like sweep and can turn twice
  • Exponential, written y = akˣ + b with x in the power, climbs or falls ever more sharply and flattens towards a horizontal line
  • The sign of the leading number reflects each shape vertically, so a negative cubic falls where a positive one rises
Ten sketches showing each standard shape and its negative twin: y = x a straight line and y = −x its mirror image, y = x² a u-shape and y = −x² an n-shape, y = x³ rising through the origin with y = −x³ falling, y = 1/x with branches in opposite quadrants and y = −1/x with the branches swapped, and y = kˣ climbing to the right against y = k⁻ˣ falling to the right
Source: Types of Graphs by Save My Exams

Fractional powers

  • Two further shapes appear because this specification allows powers of ½ and −½
  • The power ½ gives y = √x, a curve that starts at the origin and rises ever more gently
  • The power −½ gives y = 1 ÷ √x, a curve that drops steeply away from the y-axis and flattens towards the x-axis
  • Neither exists for negative x, so both curves live entirely to the right of the y-axis
Two sketches on separate axes: y = x to the power one half rising gently from the origin, and y = x to the power minus one half falling steeply from near the y-axis and flattening towards the x-axis.
Source: Types of Graphs by Save My Exams
Worked example

Matching an equation to its shape

Say what shape each of these graphs has: (a) y = 2x + 3 (b) y = x² − 4 (c) y = 6/x (d) y = 3ˣ

Solution:

  • Read the highest power of x first, because that is what fixes the family
  • (a) The highest power is 1, so it is a straight line, sloping up with gradient 2
  • (b) The highest power is 2, so it is a parabola, u-shaped because the number in front of x² is positive
  • (c) x is on the bottom of a fraction, so it is a reciprocal graph: two separate branches, one in each of two opposite quadrants
  • (d) x is in the index, not the base, so it is an exponential graph: it never touches the x-axis and climbs ever more steeply
  • The difference between (b) and (d) is exactly which position x occupies, and reading y = 3ˣ as a cubic is the standard slip