Mathematics (Core)Exam code: 0580

Further Graphs

Coordinate Geometry and Graphs

PractiseFlashcards

Recognising the Standard Shapes

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 as a vertical line x = k, is a straight line
  • Quadratic, written y = ax² + bx + c, is a single u-shape or n-shape, symmetrical about a vertical line through its lowest or highest point
  • Reciprocal, written y = a ÷ x, is two separate branches that never meet either axis
  • The sign of the number in front reflects each shape vertically, so y = −x² is an n-shape where y = x² is a u-shape
The two quadratic shapes side by side: with a greater than 0 the curve is a u-shape opening upwards with a minimum point at the bottom, and with a less than 0 it is an n-shape opening downwards with a maximum point at the top
Source: Quadratic graphs by Save My Exams
Two sketches of y = a/x showing what the value of a does: with a far from zero the branches sit well away from the axes, and with a close to zero they are pulled in tight against both axes
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 = 5 − x²

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) The highest power is again 2, so this is a parabola too, but the number in front of x² is negative, so it is n-shaped
  • Comparing (b) and (d) shows that the sign in front of the squared term, not the rest of the equation, decides which way up the curve sits

Build on this topic

Check your understanding14 exam questions, marked instantlyTest your recall13 flashcards on the key facts