Practice Maths

Graphs of Relations — Circles and Parabolas

Key Terms

These are relations (not functions), because they fail the vertical line test — except the square root function y = a√(x − h) + k which is a function.
A circle centred at the origin: x² + y² = r², radius r.
A circle with centre (h, k): (x − h)² + (y − k)² = r².
A horizontal parabola: y² = x opens to the right; y² = −x opens to the left. This is NOT a function.
The square root function: y = a√(x − h) + k is the upper (a > 0) or lower (a < 0) half of a horizontal parabola.
RelationKey features
x² + y² = r²Circle, centre (0,0), radius r
(x−h)² + (y−k)² = r²Circle, centre (h,k), radius r
y² = xHorizontal parabola, vertex (0,0), axis y=0, opens right
y = √(x − h) + kTop half of horizontal parabola; domain x ≥ h, range y ≥ k
y = −√(x − h) + kBottom half of horizontal parabola; domain x ≥ h, range y ≤ k

Circle: x² + y² = 25

x y (0,0) (5,0) (−5,0) (0,5) (0,−5) x²+y²=25

Horizontal parabola: y² = x

x y vertex (0,0) y² = x opens right
Hot Tip To find the centre and radius from a circle equation, complete the square on both x and y. For example, x² + 6x + y² − 4y = 12 becomes (x+3)² + (y−2)² = 25, so centre (−3, 2) and radius 5.

Worked Example 1 — Identifying circle features

Question: State the centre and radius of (x − 2)² + (y + 3)² = 49.

Rewrite as (x − 2)² + (y − (−3))² = 7².

Centre: (2, −3)   Radius: 7

Worked Example 2 — Complete the square to identify a circle

Question: Write x² − 4x + y² + 6y − 3 = 0 in standard form. State the centre and radius.

Group and complete the square: (x² − 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9

(x − 2)² + (y + 3)² = 16

Centre: (2, −3)   Radius: 4

Worked Example 3 — Horizontal parabola features

Question: For y² = 4x, identify the vertex, axis of symmetry, direction of opening, and state whether it is a function.

Vertex: (0, 0)

Axis of symmetry: y = 0 (the x-axis)

Opens: to the right (since coefficient of x is positive)

Function? No — for x = 4, y = ±4 (two outputs). Fails the vertical line test.

The Equation of a Circle: A Distance Argument

The equation (x − h)² + (y − k)² = r² is not a formula to memorise — it is a direct consequence of the distance formula. A circle is defined as all points exactly distance r from the centre (h, k). The distance from any point (x, y) to (h, k) is √((x−h)² + (y−k)²). Setting this equal to r and squaring both sides gives the equation of the circle. Every single point on the circle satisfies this equation; every point inside or outside does not.

This derivation shows why r appears squared: the equation comes from squaring the distance formula. The radius in the equation is r², not r, so if the equation says = 25, the radius is 5 (not 25). This is a common and costly error.

A Circle Is NOT a Function

The circle x² + y² = r² fails the vertical line test for any vertical line x = a where |a| < r. Such a line intersects the circle at two points: (a, √(r²−a²)) and (a, −√(r²−a²)). One x-value maps to two y-values, so the circle is a relation, not a function.

However, we can split a circle into two functions: the upper semicircle y = √(r²−x²) (domain −r ≤ x ≤ r, range 0 ≤ y ≤ r) and the lower semicircle y = −√(r²−x²) (range −r ≤ y ≤ 0). Each semicircle IS a function. This decomposition is important in calculus when we integrate areas bounded by circles.

Completing the Square to Find Centre and Radius

When a circle equation is given in expanded form x² + y² + Dx + Ey + F = 0, complete the square separately for the x terms and the y terms. Group: (x² + Dx) + (y² + Ey) = −F. Then add (D/2)² to both sides and (E/2)² to both sides: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F. The centre is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F).

For example: x² + y² − 6x + 4y − 12 = 0. Group: (x²−6x) + (y²+4y) = 12. Add 9 and 4: (x−3)² + (y+2)² = 25. Centre (3, −2), radius 5.

Horizontal Parabolas: Opening Left or Right

When x and y are swapped relative to a standard parabola, the curve opens horizontally. x = ay² + by + c is a horizontal parabola: it opens to the right if a > 0 and to the left if a < 0. The axis of symmetry is horizontal: y = −b/(2a). The vertex is the leftmost or rightmost point.

Horizontal parabolas are NOT functions: a vertical line through the opening region intersects the curve twice. To write them as functions, split into upper half y ≥ vertex and lower half y ≤ vertex, each giving a square root function. This is the connection between the square root function lesson and this one: y = √x is the upper half of x = y², which is the simplest horizontal parabola.

Sketching Strategy for Relations

For a circle: identify the centre by reading (h, k) from the equation (after completing the square if necessary), then plot the centre and mark points at distance r in four directions. Draw a smooth circle through them. For a horizontal parabola: find the vertex (h, k) using completing the square on the y terms (or reading it from vertex form), determine direction from the sign of a, find y-intercepts (set x = 0) and x-intercept (set y = 0). Always indicate that these are relations by noting they fail the vertical line test.

Exam Tip: In x² + y² = r², the number on the right side is r² (r squared), so the radius is √(right side). If the equation is x² + y² = 49, the radius is 7, not 49. Also: x² + y² = r² is centred at the ORIGIN — there is no (x−h) or (y−k) shifting, so the centre is (0, 0).
Exam Tip: When completing the square to find the centre and radius, you must add the same constant to BOTH sides of the equation — not just to the left side. If you add 9 to complete the square in x, the right side also increases by 9. Students frequently forget to update the right side, leading to an incorrect radius.

Mastery Practice

See Answers ➔
  1. Fluency

    State the centre and radius of each circle.

    1. (a) x² + y² = 36
    2. (b) (x − 1)² + (y − 4)² = 9
    3. (c) (x + 3)² + (y − 2)² = 25
    4. (d) x² + (y + 5)² = 100
  2. Fluency

    Write the equation of each circle.

    1. (a) Centre (0, 0), radius 8
    2. (b) Centre (3, −1), radius 5
    3. (c) Centre (−2, 4), radius √7
  3. Fluency

    For the horizontal parabola y² = x, answer the following.

    1. (a) State the vertex and direction of opening.
    2. (b) Find the y-values when x = 9.
    3. (c) Explain why y² = x is not a function but y = √x is.
    4. (d) Write the equation of the upper half of y² = 4x as a function.
  4. Fluency

    For y = √(x − 2) + 1, state the domain, range, and the endpoint of the graph.

  5. Understanding

    Complete the square to write each equation in standard circle form, then state the centre and radius.

    Method: Group x and y terms, then complete the square on each group separately.
    1. (a) x² + y² − 6x + 2y = 6
    2. (b) x² + y² + 4x − 8y + 11 = 0
  6. Understanding

    Determine whether each point lies inside, on, or outside the circle (x − 1)² + (y + 2)² = 25.

    1. (a) (4, 2)
    2. (b) (−3, −2)
    3. (c) (0, 0)
  7. Understanding

    Circles and intersections.

    Combining skills.
    1. (a) Find the x-intercepts of the circle x² + y² = 25 (set y = 0).
    2. (b) Find the y-intercepts of (x − 3)² + y² = 16 (set x = 0). State exact values.
    3. (c) Does the circle x² + y² = 4 intersect the line y = x + 3? Solve algebraically and interpret the discriminant.
  8. Understanding

    Sketching square root functions.

    1. (a) Sketch y = √x and y = −√x on the same axes. What combined shape do they form?
    2. (b) Describe the transformation from y = √x to y = 2√(x − 3) + 1. State domain and range.
  9. Problem Solving

    Equation of a circle from conditions.

    Challenge.
    1. (a) A circle has centre (2, −1) and passes through the point (5, 3). Find the radius (exact) and write the equation of the circle.
    2. (b) A circle has diameter endpoints at (1, 2) and (7, 8). Find the centre and radius, then write the equation.
    3. (c) A circle with centre (3, 0) is tangent to the y-axis. Write the equation of the circle and state the coordinates of the point of tangency.
  10. Problem Solving

    Modelling with circles.

    Challenge. A circular tunnel is modelled by the equation x² + (y − 4)² = 25, where x and y are in metres. The road through the tunnel runs along the x-axis.
    1. (a) State the centre and radius of the tunnel’s cross-section.
    2. (b) Find the width of the tunnel at road level (the x-axis, where y = 0). Give your answer in exact form.
    3. (c) A truck is 2.8 m wide and 4 m tall. Will it fit through the tunnel? Justify by checking whether the relevant points are inside the circle.