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Identify and interpret parabola features

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Rule or data
\(\displaystyle y=x^2\)

For the reference parabola, what output is paired with x = −2?

Year 10, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Rules, transformations and features

For y=a(x−h)²+k, a must be nonzero. A fixed-scale graph helps distinguish a changed multiplier from a changed viewing window

\[y=a(x-h)^2+k\]
  1. Vertex and axis The bracket vanishes at x=h, giving vertex (h,k) and vertical axis x=h
  2. Direction and width Positive a gives a minimum, negative a a maximum. Relative to the same vertex, |a|>1 is narrower and 0<|a|<1 wider
  3. Transform in a clear order Apply vertical reflection/scaling to y=x² first, then translate horizontally by h and vertically by k
  4. Intercepts and range Use y=0 for horizontal intercepts and x=0 for the vertical intercept. An upward parabola has y≥k; a downward parabola has y≤k
  5. Connect forms Expand vertex form to check an equivalent polynomial; substitute points to test either form

Worked example

Find the features of y=−2(x−1)²+8 and describe its transformation from y=x²

\[y=-2(x-1)^2+8\]
  1. \(\displaystyle V=(1,8),\quad x=1\)The vertex is a maximum and the axis is vertical
  2. \(\displaystyle y(0)=-2+8=6\)The y-intercept is (0,6)
  3. \(\displaystyle (x-1)^2=4\)Set y=0, subtract 8 and divide by −2
  4. \(\displaystyle x=-1\text{ or }x=3\)The x-intercepts are (−1,0) and (3,0)
  5. \(\displaystyle y\le8\)The domain is all real inputs. Reflect in the x-axis and stretch vertically by 2, then translate right 1 and up 8

Common mistakes

Reading the bracket sign directly
✕\(\displaystyle h=2\)✓\(\displaystyle h=-2\)
In y=(x+2)²−3, the bracket vanishes at x=−2
Writing a horizontal symmetry line
✕\(\displaystyle y=h\)✓\(\displaystyle x=h\)
The symmetry axis through (h,k) is vertical
Mixing intercept roles
✕\(\displaystyle x=0\text{ for x-intercepts}\)✓\(\displaystyle y=0\text{ for x-intercepts}\)
Every point on the horizontal axis has vertical coordinate zero
Discarding a real root
✕\(\displaystyle x=5\)✓\(\displaystyle x=1\text{ or }x=5\)
For y=(x−3)²−4, both sides of the symmetric curve cross y=0

Difficulty levels

introRead marked graph features and intercept counts
standardCalculate features from vertex-form rules
challengeDescribe reflection/dilation and calculate related features
examChoose complete feature sets or ordered transformations

Questions about this topic

Why is the bracket sign opposite the horizontal shift?+

The vertex occurs where the bracket is zero. For x+2, that happens at −2

Is the vertex always an intercept?+

No. It lies on an axis only when its corresponding coordinate is zero

Can there be one x-intercept?+

Yes. When the vertex lies on the x-axis the two equal algebraic roots describe one distinct intercept

Does stretching change the vertex?+

For y=a(x−h)²+k, changing a while holding h and k fixed leaves the vertex unchanged

Can I swap the order of transformations?+

Not always. A vertical translation followed by scaling or reflection usually differs from applying that translation last

Does a finite graph window change the domain?+

No. It only shows part of the curve unless a restricted domain is explicitly stated

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs

In this chapter

Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs