0580

Quadratic Graphs

Coordinate Geometry and Graphs

What a sketch must show

  • A sketch is not a plotted graph: it needs the right shape and the key points labelled, not accurate scales
  • Work through the same four things every time
    • Decide u-shape or n-shape from the sign of a
    • Mark the y-axis crossing at (0, c)
    • Solve for the x-axis crossings, if there are any, and mark them
    • Mark the turning point if the question asks for it
  • Draw the curve through those points, keeping it symmetrical about the vertical line through the turning point
Exam tip

Draw the curve freehand. A quadratic drawn with a ruler loses the accuracy mark, and an unlabelled curve of the right shape scores little on its own: write the coordinates against every crossing and turning point you have found.

Worked example

A curve that crosses the *x*-axis twice

Sketch the graph of y = x² − 5x + 6, showing clearly where it meets the axes.

Solution:

  • The number in front of x² is positive, so the curve is a u-shape
  • The constant term is the y-axis crossing, so the curve passes through (0, 6)
  • For the x-axis crossings, solve x² − 5x + 6 = 0
  • Two numbers multiplying to 6 and adding to −5 are −2 and −3, so (x − 2)(x − 3) = 0
  • The curve meets the x-axis at (2, 0) and (3, 0)
  • Draw a u-shaped curve through the three marked points
A sketched u-shaped quadratic curve crossing the y-axis at (0, 6) and the x-axis at (2, 0) and (3, 0), each of the three points marked with a cross and labelled with its coordinates
Source: Quadratic graphs by Save My Exams
Worked example

A curve that never meets the *x*-axis

Sketch the graph of y = x² − 6x + 13, showing the y-axis crossing and the coordinates of the turning point.

Solution:

  • The number in front of x² is positive, so the curve is a u-shape and its turning point is a minimum
  • The y-axis crossing is at (0, 13)
  • Complete the square to find the turning point
  • Half of −6 is −3, so x² − 6x is (x − 3)² − 9
  • y = (x − 3)² − 9 + 13, which is y = (x − 3)² + 4
  • The turning point is at (3, 4)
  • That minimum sits above the x-axis on a u-shaped curve, so the curve never crosses the x-axis at all
A sketched u-shaped quadratic curve crossing the y-axis at (0, 13) with its minimum point at (3, 4) marked, the whole curve lying above the x-axis so it never crosses it
Source: Quadratic graphs by Save My Exams
Worked example

A curve that touches the *x*-axis once

Sketch the graph of y = −x² − 4x − 4, showing the root, the y-axis crossing and the turning point.

Solution:

  • The number in front of x² is negative, so the curve is an n-shape with a maximum
  • The y-axis crossing is at (0, −4)
  • Factorise: −x² − 4x − 4 = −(x² + 4x + 4) = −(x + 2)²
  • Setting y = 0 gives (x + 2)² = 0, so x = −2 is the only solution
  • A repeated solution means the curve touches the x-axis rather than cutting through it, so (−2, 0) is both the root and the turning point
A sketched n-shaped quadratic curve whose highest point just touches the x-axis at (−2, 0), falling away to cross the y-axis at (0, −4), both points marked with a cross and labelled
Source: Quadratic graphs by Save My Exams