Graphs of Non-Linear FunctionsEdexcel GCSE Maths: Revision notes

Non-linear functions produce curved graphs rather than straight lines. Understanding the key features of each type is essential for sketching and interpreting graphs.

Quadratic functions (y = ax² + bx + c):

• Produce a parabola shape
• Always have a turning point (minimum if a > 0, maximum if a < 0)
• Are symmetrical about a vertical line through the turning point
• Cross the y-axis at (0, c)

Cubic functions (y = ax³ + bx² + cx + d):

• Produce an S-shaped curve
• Can have up to two turning points
• As x → ∞, y → ∞ (if a > 0) or y → -∞ (if a < 0)
• Always cross the x-axis at least once

Reciprocal functions (y = k/x):

• Produce two separate curves (asymptotic)
• Have a vertical asymptote at x = 0 and horizontal asymptote at y = 0
• Do not cross either axis
• In quadrant I if k > 0; in quadrants II and IV if k < 0

Exponential functions (y = a^x):

• Curve that never goes below y = 0
• Have a horizontal asymptote at y = 0
• If a > 1: curve increases rapidly; if 0 < a < 1: curve decreases
• Always pass through (0, 1)

Quadratic graphs have distinct features that can be found algebraically or graphically.

Roots (x-intercepts):

• Points where the graph crosses the x-axis (where y = 0)
• Found by solving ax² + bx + c = 0
• A quadratic can have 0, 1, or 2 real roots

Turning point (vertex):

• The maximum or minimum point of the parabola
• x-coordinate: x = -b/(2a)
• y-coordinate: substitute the x-value back into the equation
• Minimum turning point if a > 0; maximum if a < 0

Line of symmetry:

• A vertical line through the turning point
• Equation: x = -b/(2a)
• The parabola is mirror-symmetric about this line

y-intercept:

• Where the graph crosses the y-axis (where x = 0)
• Always at the point (0, c) in y = ax² + bx + c

Completing the square can also reveal the turning point directly:

• y = a(x + p)² + q gives turning point at (-p, q)

Graphs can be used to solve equations that are difficult or impossible to solve algebraically.

Method for solving f(x) = k graphically:

1. Draw the graph of y = f(x)
2. Draw the horizontal line y = k
3. Find the point(s) where the graphs intersect
4. Read off the x-coordinate(s) of the intersection point(s)

Method for solving f(x) = g(x) graphically:

1. Draw both graphs on the same axes
2. Identify intersection point(s)
3. Read the x-coordinate(s) of intersection points

Accuracy:

• Solutions found graphically are approximate unless they coincide exactly with grid intersections
• State answers to appropriate precision (usually 1 or 2 decimal places)
• The finer the scale of your graph, the more accurate your solution

Common applications:

• Solving quadratic equations: find where y = ax² + bx + c crosses y = 0 (the x-axis)
• Comparing growth rates: find where exponential and linear graphs intersect
• Finding break-even points in real-world contexts

Graphical solutions are particularly useful when algebraic methods are time-consuming or when dealing with combinations of different function types.

Sine, cosine and tangent functions produce distinctive periodic curves.

y = sin x:

• Oscillates between -1 and +1
• Period: 360° (or 2π radians)
• Crosses y = 0 at x = 0°, 180°, 360°, ...
• Maximum at 90°; minimum at 270°
• Smooth, S-shaped waves

y = cos x:

• Oscillates between -1 and +1
• Period: 360° (or 2π radians)
• Crosses y = 0 at x = 90°, 270°, ...
• Maximum at 0°, 360°, ...; minimum at 180°
• Similar to sine but shifted 90° to the left

y = tan x:

• Unbounded (no maximum or minimum)
• Period: 180° (or π radians)
• Has vertical asymptotes at x = 90°, 270°, ...
• Crosses y = 0 at x = 0°, 180°, 360°, ...
• Discontinuous with sharp increases between asymptotes

Using trigonometric graphs:

• Identify where y = sin x = a (find x-values)
• Solve equations like sin x = 0.5 by reading from the graph
• Determine periodicity and symmetry properties

Transformations allow you to sketch variations of standard graphs without plotting many points.

Translation (shift) — f(x + a) and f(x) + a:

• f(x + a): horizontal translation LEFT by a units

  • Affects x-values only
  • y = (x + 2)² moves the parabola 2 units left
  • (x + a) means move left; (x - a) means move right

• f(x) + a: vertical translation UP by a units

  • Affects y-values only
  • y = x² + 3 moves the parabola 3 units up
  • +a means up; -a means down

Scaling (stretch) — f(ax) and af(x):

• f(ax): horizontal stretch/compression by factor 1/a

  • Affects x-coordinates
  • If a > 1: graph compresses horizontally (narrower)
  • If 0 < a < 1: graph stretches horizontally (wider)
  • y = (2x)² is half as wide as y = x²

• af(x): vertical stretch by factor a

  • Affects y-coordinates
  • If a > 1: graph stretches vertically (taller)
  • If 0 < a < 1: graph compresses vertically (shorter)
  • If a < 0: also reflects in the x-axis
  • y = 3x² is three times as tall as y = x²

Combining transformations:

• Apply transformations in the correct order: usually consider horizontal changes (a, x) before vertical changes
• y = 2(x - 1)² + 3: move right 1, stretch vertically by 2, move up 3

Circles are defined by a specific algebraic equation and have unique properties related to tangent lines.

Equation of a circle centred at the origin:

• Standard form: x² + y² = r²
• Where r is the radius
• Every point (x, y) on the circle is at distance r from the origin
• Examples: x² + y² = 9 is a circle with radius 3; x² + y² = 25 has radius 5

Key properties of circles:

• The circle is defined by all points equidistant from the centre
• To find if a point lies on the circle, substitute its coordinates and check if the equation is satisfied
• The circle passes through (r, 0), (0, r), (-r, 0), (0, -r)

Tangent to a circle at a given point:

• A tangent is a straight line that touches the circle at exactly one point
• The tangent is perpendicular to the radius at the point of contact

Method to find the equation of a tangent:

1. Find the gradient of the radius from the origin (0, 0) to the point of contact (a, b)
   • Gradient of radius = b/a
2. The tangent is perpendicular, so its gradient = -a/b (negative reciprocal)
3. Use y - b = m(x - a) where m = -a/b
4. Rearrange to the form ax + by = a² + b² (using the fact that a² + b² = r² for a point on the circle)

Alternative tangent equation:

• For a point (a, b) on circle x² + y² = r², the tangent line is: ax + by = r²

This elegant result comes directly from the perpendicularity condition.