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Read parabola features from a graph

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A labelled coordinate graph. Read the horizontal coordinate first and the vertical coordinate second.VABY-224-6-4-22460xy

Which ordered pair describes the turning point V?

Year 9, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Read the graph as a connected picture

Axes, numbered scales, plotted points and labels must agree. Check a known point before interpreting other features

\[\begin{gathered}V=(h,k)\\\text{symmetry axis: }x=h\end{gathered}\]
  1. Coordinates Read horizontal x first, then vertical y.
  2. Turning point Opening up gives a minimum; opening down gives a maximum
  3. Intercepts At horizontal-axis crossings y=0. At the vertical-axis crossing x=0.
  4. Construct Make a table, choose uniform labelled scales, plot points and draw a smooth curve. Check the result digitally

Worked example

For y=−(x+1)²+4, read or calculate the vertex, symmetry axis and intercepts

\[y=-(x+1)^2+4\]
  1. \(\displaystyle V=(-1,4)\)The squared term is zero at x=−1. The graph opens down, so 4 is a maximum
  2. \(\displaystyle x=-1\)The symmetry axis is vertical
  3. \(\displaystyle \begin{gathered}(x+1)^2=4\\x=-3\text{ or }1\end{gathered}\)Set y=0 for the x-intercepts. The x-intercepts are (−3,0) and (1,0)
  4. \(\displaystyle y(0)=3\)The y-intercept is (0,3)

Common mistakes

Swapping coordinate order
✕\(\displaystyle V=(k,h)\)✓\(\displaystyle V=(h,k)\)
Horizontal coordinate first, vertical coordinate second
Writing a horizontal symmetry line
✕\(\displaystyle y=h\)✓\(\displaystyle x=h\)
An upright parabola has a vertical symmetry axis
Counting a repeated root twice
✕\(\displaystyle \text{two distinct roots}\)✓\(\displaystyle \text{one distinct root}\)
A single touching point is one distinct x-intercept

Difficulty levels

introRead symmetry and turning type, including shifted and zero-intercept cases
standardRead vertex and intercept coordinates from an exact scaled graph
challengeCombine features, including repeated roots and non-unit vertical scales
examInterpret a graph in a bounded physical or geometric context

Questions about this topic

Does one grid gap always mean one unit?+

No. Read numbered values on each axis separately. The horizontal and vertical scales can differ

Is the axis of symmetry the y-axis?+

Only when the vertex has x-coordinate zero. Otherwise it is another vertical line x=h.

Can the vertex also be an intercept?+

Yes. When the vertex is on y=0, it is the one repeated root. At the origin it is also the y-intercept

Does changing root spacing leave the vertex height fixed?+

Not in y=(x−h)²−d². Changing |d| changes both the spacing 2|d| and the vertex height −d²

Can a parabola have no real roots?+

Yes. It has no real roots when it does not meet the x-axis

Can I read any number of decimal places from a graph?+

No. The scale limits visual precision. Use numerical evaluation or a graphing tool to refine an approximation

Does a sketch prove a physical model outside its domain?+

No. Interpret only the part supported by the context and evidence

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