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Find a quadratic rule from its roots

Start a set of 8
Step 2 of 21

Fit y=a(x+1)(x−3) through P=(1,2). The roots stay fixed. Q shows the candidate output at x=1. Change a until Q meets P, then compare the opening and height

Parabolas with roots at -1 and 3. Target point P is (1,2). Candidate point Q moves vertically with the scale; its label merges with P when the rule fits.R₁R₂PQ-224-9-6-33690xy
Required output
2
Candidate output at x=1
-4

Which scale fits P=(1,2)?

Year 9, Chapter 10, 1 difficulty level, No calculatorBuild a worksheet

Rule from roots and another point

Build one factor for each root, then find the nonzero scale using another point

\[y=a(x-r_1)(x-r_2)\]
  1. Root factors A root r supplies x−r, including x when the root is zero
  2. Find the scale Substitute both coordinates of the extra point, evaluate the brackets and divide
  3. Verify Check both roots and the extra point in the completed rule
  4. Enough information An extra point at an existing root cannot determine the scale. A repeated root must be explicitly established

Worked example

Roots are −2 and 4. The graph passes through (0,4). Find its rule

\[y=a(x+2)(x-4)\]
  1. \(\displaystyle 4=a(0+2)(0-4)\)Substitute both coordinates
  2. \(\displaystyle 4=-8a\)Multiply the brackets
  3. \(\displaystyle a=-\frac12\)Divide by −8
  4. \(\displaystyle y=-\frac12(x+2)(x-4)\)Write the complete factorised rule
  5. \(\displaystyle y(0)=-\frac12(2)(-4)=4\)The extra point fits, and each root still makes a factor zero

Common mistakes

Using the point output as scale
✕\(\displaystyle P=(0,4)\Rightarrow a=4\)✓\(\displaystyle 4=-8a\Rightarrow a=-\frac12\)
Substitute into the whole factorised rule before solving
Reversing both root signs
✕\(\displaystyle \begin{gathered}r_1=-2,\ r_2=4\\y=a(x-2)(x+4)\end{gathered}\)✓\(\displaystyle \begin{gathered}r_1=-2,\ r_2=4\\y=a(x+2)(x-4)\end{gathered}\)
Each factor must be zero at its own root
Dividing by a zero bracket product
✕\(\displaystyle 0=0a\Rightarrow a=0\)✓\(\displaystyle 0=0a\text{ for every }a\)
A point already known to be an intercept gives no new scale equation

Difficulty levels

introConstruct a factorised rule from two distinct roots and an extra point, then evaluate it

Questions about this topic

Are two roots enough?+

They fix the factors but leave the nonzero scale unknown

Where must the extra point be?+

Its x-coordinate must differ from both roots so the bracket product is nonzero. Its y-coordinate must also be nonzero for the rule to be quadratic

Can a root be zero?+

Yes. Use a factor x and keep the origin as an intercept

Can the scale be fractional or negative?+

Yes. Use the exact value from substitution; a negative value opens the parabola downward

Does root order matter?+

The product is unchanged when the factors swap. In practice fields, enter the smaller root first when requested

What about a repeated root?+

If a repeated root is explicitly given, square its factor. Use an additional point whose x-coordinate differs from that root to find the scale

What if the point contradicts a root?+

No rule can make the same input have both zero and a nonzero output. Report inconsistent information

Must I expand the final rule?+

Only when asked for expanded form. Here the task requests a factorised rule

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 10: Quadratic equations and graphs

In this chapter

Recognise, standardise and test quadratic equationsSolve common-factor and square-root quadratic equationsFactorise and solve monic quadratic equationsFormulate and solve a quadratic problemAnalyse and optimise a quadratic modelRead parabola features from a graphSolve quadratics graphically and numerically10F Parabola transformations: Transform a quadratic and find its key features10G Turning-point form: Connect turning-point and expanded forms10G Turning-point form: Find a quadratic rule from its turning point10H Parabola intercepts and features: Connect factors with parabola features10H Parabola intercepts and features: Find a quadratic rule from its rootsInvestigate related function graphs