National 5MathsSQARevision NotesAlgebraic SkillsQuadratic FunctionsEquation of a Quadratic Graph

Equation of a Quadratic Graph (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Determining the equation of a quadratic from its graph

How do I determine the equation of a quadratic in the form y = kx2 from its graph?

  • The graph of a quadratic with equation bold italic y bold equals bold italic k bold italic x to the power of bold 2 will always have its turning point at the origin (0, 0)

    • x equals 0 will be its axis of symmetry

    • If k is positive it will be an 'up' or union-shaped parabola

    • If k is negative it will be a 'down' or intersection-shaped parabola

  • If you are given a quadratic graph with its turning point at the origin and need to find its equation

    • The equation will be of the form y equals k x squared

    • You just need to find the value of k by using the coordinates of another point on the graph

  • For example, if a quadratic graph has its turning point at the origin and goes through the point open parentheses 2 comma space minus 12 close parentheses

    • Substitute x equals 2 and y equals negative 12 into y equals k x squared

      • negative 12 equals k open parentheses 2 close parentheses squared

    • And solve to find the value of k

      • negative 12 equals 4 k space space rightwards double arrow space space k equals negative 3

    • The equation is y equals negative 3 k squared

How do I determine the equation of a quadratic in the form y = k(x+a)2+b from its graph?

  • The graph of a quadratic with its equation in completed square form Error converting from MathML to accessible text. will have its turning point at (-a, b)

    • x equals negative a will be its axis of symmetry

    • If k is positive it will be an 'up' or union-shaped parabola

    • If k is negative it will be a 'down' or intersection-shaped parabola

  • If you are given a quadratic graph with its turning point not at the origin and need to find its equation

    • The equation will be of the form y equals k stretchy left parenthesis x plus a stretchy right parenthesis squared plus b

    • You can get the values of space a and b from the coordinates of the turning point (-a, b)

      • Remember to switch the sign of the x-coordinate to get a

    • You can find the value of k by using the coordinates of another point on the graph

      • If that other point is the space bold italic y-intercept remember that that point has an x-coordinate of 0

  • For example, if a quadratic graph has its turning point at open parentheses negative 2 comma space 3 close parentheses and goes through the point open parentheses 1 comma space 21 close parentheses

    • From the turning point, space a equals 2 and b equals 3

      • y equals k open parentheses x plus 2 close parentheses squared plus 3

    • From the point open parentheses 1 comma space 21 close parentheses, substitute x equals 1 and y equals 21 into y equals k open parentheses x plus 2 close parentheses squared plus 3

      • 21 equals k open parentheses 1 plus 2 close parentheses squared plus 3

    • And solve to find the value of k

      • 21 equals 9 k plus 3 space space rightwards double arrow space space 9 k equals 18 space space rightwards double arrow space space k equals 2

    • The equation is y equals 2 open parentheses x plus 2 close parentheses squared plus 3

  • If you have found the values of k comma space a and bin y equals k stretchy left parenthesis x plus a stretchy right parenthesis squared plus b, and need to find the coordinates of the space bold italic y-intercept

    • Remember that the y-intercept has coordinates open parentheses 0 comma space c close parentheses for some value of c

    • Substitute x equals 0 into the equation and solve to find the value of c

    • For example, for y equals 2 open parentheses x plus 2 close parentheses squared plus 3

      • y equals 2 open parentheses 0 plus 2 close parentheses squared plus 3 equals 2 open parentheses 4 close parentheses plus 3 equals 8 plus 3 equals 11

      • The y-intercept is at open parentheses 0 comma space 11 close parentheses

Examiner Tips and Tricks

Be careful with the two different forms of a quadratic, y equals a x squared plus b x plus c and y equals k stretchy left parenthesis x plus a stretchy right parenthesis squared plus b.

  • y equals a x squared plus b x plus c has its y-intercept at open parentheses 0 comma space c close parentheses

  • But y equals k stretchy left parenthesis x plus a stretchy right parenthesis squared plus b does not have its y-intercept at open parentheses 0 comma space b close parentheses

Worked Example

The graph below shows part of a parabola of the form y equals open parentheses x plus a close parentheses squared plus b.

Graph of a parabola opening upwards with vertex at (4, 3), intersecting the y-axis at point P, on a coordinate plane with x and y axes.

(a) Write down the equation of the axis of symmetry of the graph.

(b) State the values of a and b.

(c) straight P is the point open parentheses 0 comma space c close parentheses.  Find the value of c.

Answer:

Part (a)

The axis of symmetry of a quadratic always goes through its turning point

x equals 4

Part (b)

A quadratic with equation in the form y equals open parentheses x plus a close parentheses squared plus b will have its turning point at open parentheses negative a comma space b close parentheses

a equals negative 4 comma space space b equals 3

Part (c)

The equation of the parabola is y equals open parentheses x minus 4 close parentheses squared plus 3

The curve goes through the point open parentheses 0 comma space c close parentheses, so y equals c when x equals 0

  • Substitute those values into the equation and solve for c

table row c equals cell open parentheses 0 minus 4 close parentheses squared plus 3 end cell row blank equals cell 16 plus 3 end cell row blank equals 19 end table

c equals 19

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Previous:Features of Quadratic GraphsNext:Sketching Quadratics
    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.