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Connect factors with parabola features

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Step 1 of 22

For y=(x+2)(x−4), which x-values make the output zero?

Year 9, Chapter 10, 3 difficulty levels, No calculatorBuild a worksheet

Features from factors

Use each form for the feature it reveals, then check that all forms describe the same quadratic

\[\begin{gathered}y=a(x-r_1)(x-r_2)\\a\ne0\end{gathered}\]
  1. X-intercepts Set each factor to zero. Report each distinct point (r,0)
  2. Symmetry The axis x-value is the average of the two roots, including a repeated pair
  3. Vertex Substitute the axis x-value into the rule to obtain the height
  4. Y-intercept Substitute x=0. Do not confuse this height with the vertex height
  5. Opening A positive leading scale opens upward; a negative scale opens downward

Worked example

Find the roots, axis, vertex and y-intercept

\[y=-(x+1)(x-5)\]
  1. \(\displaystyle x=-1\text{ or }x=5\)Set each variable factor to zero
  2. \(\displaystyle h=\frac{-1+5}{2}=2\)Average the roots
  3. \(\displaystyle y(2)=-(3)(-3)=9\)The vertex is (2,9), a maximum
  4. \(\displaystyle y(0)=-(1)(-5)=5\)The vertical-axis intercept is (0,5)
  5. \(\displaystyle \begin{gathered}y=-(x-2)^2+9\end{gathered}\)Expansion confirms this equivalent turning-point form

Common mistakes

Reading the factor sign
✕\(\displaystyle x+3=0\Rightarrow x=3\)✓\(\displaystyle x+3=0\Rightarrow x=-3\)
Solve the factor equation instead of copying its sign
Confusing the intercept axes
✕\(\displaystyle x=4\Rightarrow(0,4)\)✓\(\displaystyle x=4\Rightarrow(4,0)\)
A root has zero output
Losing a root by division
✕\(\displaystyle x(x-4)=0\Rightarrow x=4\)✓\(\displaystyle \begin{gathered}x(x-4)=0\\\Rightarrow x=0\text{ or }4\end{gathered}\)
Treat the x=0 possibility before dividing by x.

Difficulty levels

standardConnect factorised and turning-point features
challengeCombine features with exact table values
examCombine transformations, coordinates, roots and evaluation

Questions about this topic

Why average the roots?+

Two distinct roots lie symmetrically on either side of the axis of symmetry. Their midpoint has the vertex x-coordinate

What if both roots are the same?+

There is one distinct intercept, and it is also the vertex

Can one root be zero?+

Yes. The rule has a factor x, and the graph passes through the origin

Does changing a move the roots?+

For nonzero a, no. It changes other heights and may change the opening

What if a is zero?+

The rule becomes y=0, so it is a line rather than a quadratic

Is the vertex height the y-intercept?+

Only when the vertex is on the y-axis. Calculate them at their respective x-values

Do I have to expand to read roots?+

No. Factorised form already shows them. Expansion can help connect another equivalent form

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