Vertex/intercept/axis features | Year 11

01 / LEARN

The idea, step by step

Read square form

For $y = a {(x - h)}^{2} + k$ , the vertex is (h,k) and the symmetry line is x=h. Positive a opens upward; negative a opens downward. A larger absolute a gives a narrower curve at a fixed height from the vertex.

The domain is all real x unless restricted. For upward opening, the range starts at k. For downward opening, it ends at k.

y = a {(x - h)}^{2} + k

Calculate the intercepts

Put x=0 for the y-intercept and y=0 for the x-intercepts. If roots are r and s, their midpoint gives the axis x=(r+s)/2.

Label the vertex, axes and intercepts. One extra symmetric pair of points helps confirm the shape without requiring a huge table.

A common trap

Reading ${(x + 4)}^{2} - 1$ as vertex (4,-1).

Instead: The bracket is zero at x=-4, so the vertex is (-4,-1).

02 / WORKED EXAMPLES

See the thinking

Example 1

A shifted minimum

State vertex, axis and range of $y = 2 {(x - 3)}^{2} - 5$ .

The squared bracket is zero at x=3.
The leading coefficient is positive, so the minimum is -5.

Vertex (3,-5); axis x=3; range

y \geq - 5

Example 2

Features from factors

Find intercepts and vertex of $y = x^{2} - 4 x$ .

Factor $y = x (x - 4)$ , giving roots 0 and 4.
The axis is their midpoint x=2.
Substitute x=2 to obtain y=-4.

Intercepts (0,0),(4,0); vertex (2,-4).

Check your understanding

Find the vertex of $y = - 3 {(x + 2)}^{2} + 7$ .

Show answer

(-2,7), a maximum.

The bracket vanishes at -2 and the coefficient is negative.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State vertex/axis for $y = {(x - 5)}^{2} + 2$ .
2 marks
Find the y-intercept of $y = - 2 {(x + 1)}^{2} + 8$ .
2 marks
Find x-intercepts of $y = {(x - 1)}^{2} - 9$ .
2 marks
Find the vertex of $y = x^{2} + 6 x + 4$ .
2 marks
Sketch $y = - {(x - 2)}^{2} + 4$ , labelling vertex/intercepts.
4 marks
Give domain/range of $y = 3 {(x + 1)}^{2} - 6$ for real x.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

State vertex/axis for $y = {(x - 5)}^{2} + 2$ .
2 marks
Read square form.
Vertex (5,2); axis x=5.
Find the y-intercept of $y = - 2 {(x + 1)}^{2} + 8$ .
2 marks
At x=0, y=-2+8=6.
(0,6)
Find x-intercepts of $y = {(x - 1)}^{2} - 9$ .
2 marks
Set ${(x - 1)}^{2} = 9$ .
Then x-1 is plus/minus 3.
(-2,0),(4,0)
Find the vertex of $y = x^{2} + 6 x + 4$ .
2 marks
Rewrite as ${(x + 3)}^{2} - 5$ .
(-3,-5)
Sketch $y = - {(x - 2)}^{2} + 4$ , labelling vertex/intercepts.
4 marks
Solve ${(x - 2)}^{2} = 4$ .
Use symmetry x=2.
Vertex (2,4); roots (0,0),(4,0); opens downward.
Give domain/range of $y = 3 {(x + 1)}^{2} - 6$ for real x.
2 marks
The square is nonnegative.
Its minimum is zero at x=-1.
Domain all real numbers; range $y \geq - 6$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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