Grade 11 Functions and Graphs

Grade 11 functions: parabola in turning-point form, hyperbola y = a/(x + p) + q, exponential y = a·bˣ⁺ᵖ + q, average gradient and intersections, with…

Functions and graphs are about 45 of the 150 marks in Grade 11 Paper 1. The big new idea is the horizontal shift p: the parabola y = a(x + p)² + q, the hyperbola y = a/(x + p) + q and the exponential graph y = a·bˣ⁺ᵖ + q.

You sketch these graphs from their equations, find equations from sketches, work out intercepts, turning points, axes of symmetry and asymptotes, find points of intersection, and calculate the average gradient between two points. The trig graphs with horizontal shifts and period changes are examined in Paper 2.

Key concepts

  • Parabola y = a(x + p)² + q: turning point (−p; q), axis of symmetry x = −p
  • Parabola y = ax² + bx + c: x = −b/2a for the axis of symmetry
  • Hyperbola y = a/(x + p) + q: asymptotes x = −p and y = q
  • Exponential y = a·bˣ⁺ᵖ + q: asymptote y = q
  • Average gradient between two points: [f(x₂) − f(x₁)] / (x₂ − x₁)
  • Points of intersection by solving f(x) = g(x)

Worked examples

Write down the turning point and range of y = −2(x + 1)² + 8.

The turning point is (−p; q) = (−1; 8). a < 0, so the parabola opens downwards and the range is y ≤ 8.

Write down the domain and asymptotes of y = 3/(x − 2) + 1.

The asymptotes are x = 2 and y = 1. The domain is x ∈ ℝ, x ≠ 2.

Find the x-intercepts of f(x) = x² − 2x − 8.

Solve x² − 2x − 8 = 0: (x − 4)(x + 2) = 0, so the x-intercepts are (4; 0) and (−2; 0). The axis of symmetry is halfway between them, x = 1.

Find the average gradient of f(x) = x² between x = 1 and x = 3.

f(3) = 9 and f(1) = 1. Average gradient = (9 − 1)/(3 − 1) = 4.

Common mistakes

  • Writing the turning point of y = (x + 1)² + 8 as (1; 8) instead of (−1; 8).
  • Drawing a hyperbola that touches its asymptotes.
  • Confusing the average gradient with the gradient at a point (which is Grade 12 calculus).
  • Reading intersections off a sketch instead of solving algebraically when the question says 'calculate'.

Exam tips

  • The sign inside the bracket is the opposite of the x-value of the turning point.
  • Find a by substituting a point on the graph that isn't the turning point.
  • Label every intercept, turning point and asymptote on a sketch: that is where the marks are.

Frequently asked questions

What is new in Grade 11 functions compared with Grade 10?

The horizontal shift p. Grade 10 graphs are only shifted up and down (q); Grade 11 adds left and right shifts, and the parabola in turning-point form.

Do I need calculus for turning points in Grade 11?

No. Use completing the square or x = −b/2a. Calculus for turning points and gradients at a point is Grade 12.

Are inverse functions in Grade 11?

No. Inverse functions and logarithmic graphs are Grade 12 work.

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