MathematicsExam code: 0580

Further Graphs and Tangents

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 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 y-axis, and y = a ÷ x never meets the x-axis either
  • 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

Worked example

Where a cubic meets the axes

(a) Sketch the graph of y = (x − 2)(x + 1)², showing the values where it meets the axes.

(b) Find the exact x-coordinates of the three points where the graph of y = x³ − 6x crosses the x-axis.

Solution:

  • (a) y = 0 when a factor is zero: x − 2 = 0 gives x = 2, and x + 1 = 0 gives x = −1
  • The factor (x + 1) is squared, so x = −1 is a repeated root: the curve touches the x-axis there and turns back instead of crossing it
  • At x = 0, y = (0 − 2) × (0 + 1)² = −2, so the curve cuts the y-axis at (0, −2)
  • Multiplied out, the first term is +x³, so this is a positive cubic: it rises from the bottom left and leaves at the top right
  • So the curve climbs to touch the axis at (−1, 0), falls through (0, −2) to a minimum, then rises to cross the axis at (2, 0)
  • Label −1 and 2 on the x-axis and −2 on the y-axis
A sketch of y = (x − 2)(x + 1)² on plain axes: the curve rises from the bottom left to touch the x-axis at −1 and turn back, falls through −2 on the y-axis to a minimum, then rises to cross the x-axis at 2 and continues up to the top right
  • (b) Take out the common factor x: x³ − 6x = x(x² − 6) = 0
  • Either x = 0, or x² = 6, which gives x = √6 or x = −√6
  • The crossing points have x-coordinates −√6, 0 and √6, left exact as the question asks

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