Maths · Algebra / Precalculus · suggested level; school sequences vary

Quadratic Graphs: Vertex, Intercepts & Sketching Examples

Two different x-values can land at the same height on a parabola. Find its centre of symmetry first, and the rest of the sketch becomes easier to check.

Start here

For y=a(x−h)²+k, the vertex is (h,k) and the axis is x=h. Use intercepts and symmetric point pairs to check the graph.

Original BananaLearn guide · 7 November 2026

Sketch from the vertex and symmetric pairs
Sketch from the vertex and symmetric pairs. Open full-size illustration

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Find a section, worked example or practice question

What makes a graph quadratic?

A quadratic has a non-zero squared term: y=ax²+bx+c with a≠0. Its graph is a parabola. Positive a gives an upward opening; negative a gives a downward opening. The graph is not two straight lines meeting at a corner.

The vertex is the turning point. An upward-opening parabola has a minimum there; a downward-opening one has a maximum. Naming the direction helps you catch a sketch that contradicts the equation.

Read vertex form correctly

In y=a(x−h)²+k, the squared part is zero at x=h. Therefore the vertex is (h,k). For y=2(x−3)²−4, use (3,−4), not (−3,−4). The axis of symmetry is the vertical line x=3.

Points equally far to the left and right of h have the same square and therefore the same y-value. This gives an efficient table and a strong visual check.

Find the y-intercept

Set x=0, because every point on the y-axis has x-coordinate zero. For y=x²−4x+3, the y-intercept is (0,3). In general form it is (0,c).

The number c is not automatically the vertex height. A parabola can meet the y-axis well above or below its turning point. Keep the roles of intercept and vertex separate.

One quadratic, three useful forms
One quadratic, three useful forms. Open full-size illustration

On a narrow screen, swipe the diagram or open it full size.

Find the x-intercepts, if they exist

Set y=0 and solve the quadratic. The graph may cross twice, touch once or never meet the x-axis over the real numbers. For y=(x−1)(x−3), the intercepts are (1,0) and (3,0).

For y=(x−2)²+5, a square is non-negative, so y is always at least 5. There are no real x-intercepts. Do not force two crossings onto every quadratic sketch.

Move between forms for a reason

Expanded form makes the y-intercept clear. Factorised form makes roots clear when factorisation is available. Vertex form makes the turning point clear. They are three descriptions of one function, not three unrelated formulas.

Complete the square in x²−4x+3: (x−2)²−1. The vertex is (2,−1), midway between roots 1 and 3. Expanding the vertex form returns the original expression.

Build a table around the vertex

For y=(x−2)²−1, choose x=0,1,2,3,4. The y-values are 3,0,−1,0,3. Plot the vertex and the two symmetric pairs. Draw a smooth curve through them rather than connecting points with a zigzag.

Label both axis scales and the equation. A sketch is qualitative unless accurate plotting is required; it should still put crossings and the vertex on the correct sides of each axis.

Use the graph to discuss range and equations

For this upward parabola, every y≥−1 occurs and no smaller value occurs. Its range is y≥−1. A horizontal line y=3 meets it at x=0 and x=4, so the equation (x−2)²−1=3 has two solutions.

A restricted x-domain changes the displayed graph and may change the range. State any restriction before treating a sketch as a complete description.

A final sketch checklist

Check opening direction, vertex, axis, y-intercept, real roots and one symmetric pair. If these disagree, revisit the equation before improving the artwork. The coefficient a changes vertical scale as well as opening direction.

For parent support, ask “which part of the formula tells you that?” A child who can connect each feature to the equation is learning more than a memorised sequence of plotting steps.

See the reasoning

Worked examples and checks

1. Sketch y=x²−4x+3.

  1. Factorise to (x−1)(x−3): roots 1 and 3.
  2. Complete the square to (x−2)²−1: vertex (2,−1).
  3. The y-intercept is (0,3); (4,3) is its symmetric partner. Open upward.

2. Describe y=−2(x+1)²+8.

  1. The vertex is (−1,8); the graph opens downward.
  2. For roots: 2(x+1)²=8, so x+1=±2 and x=−3 or 1.
  3. At x=0, y=6. Range: y≤8.

3. Does y=(x−4)²+2 cross the x-axis?

  1. Its minimum is 2, reached at x=4.
  2. Zero is below that minimum, so there are no real roots.
  3. The graph lies above the x-axis, with axis x=4.

Try, explain, check

12 questions with explained answers

Try each task before opening its explanation. These are original BananaLearn questions, not official exam or past-paper questions. Where several answers can work, the explanation gives the criteria.

  1. Find the vertex of (x−5)²+3.

    Check answer and reasoning

    (5,3)
    The square vanishes at x=5.

  2. Find the axis of y=(x+2)²−7.

    Check answer and reasoning

    x=−2
    The centre is where x+2=0.

  3. Does y=−x²+4 open up or down?

    Check answer and reasoning

    Down
    The squared-term coefficient is negative.

  4. Find its roots.

    Check answer and reasoning

    x=−2 and x=2
    Set x²=4.

  5. Find the y-intercept of y=2x²−3x−6.

    Check answer and reasoning

    (0,−6)
    Substitute x=0.

  6. What is the range of y=3(x−1)²+2?

    Check answer and reasoning

    y≥2
    The square is non-negative and a is positive.

  7. What is the maximum of y=−(x−4)²+9?

    Check answer and reasoning

    9 at x=4
    Subtracting a square cannot increase 9.

  8. For y=(x−3)², y=4 at x=1. Find the symmetric x-value.

    Check answer and reasoning

    5
    Both are 2 units from x=3.

  9. How many real roots has y=x²+1?

    Check answer and reasoning

    None
    Its minimum is 1.

  10. How many real roots has y=(x+6)²?

    Check answer and reasoning

    One distinct root, −6
    The curve touches the axis at its vertex.

  11. Rewrite x²+6x+5 in vertex form.

    Check answer and reasoning

    (x+3)²−4
    Expand to check x²+6x+9−4.

  12. Is the vertex always the y-intercept?

    Check answer and reasoning

    No
    They coincide only when the vertex lies on x=0.

A checking record to use in your notebook

Copy these headings and fill them for one worked task, then for a fresh attempt. Use your own evidence rather than copying the model answer.

Question or sentence
What am I being asked to explain?
Model or evidence
Which diagram, rule or passage detail supports my method?
Work
What steps connect the question to the answer?
Independent check
Can I substitute, recount, trace a path or check the text?
Next attempt
What changes in a fresh example?

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Original BananaLearn explanations, examples, illustrations and practice. Source facts checked 7 November 2026. Suggested levels are learning guides, not a claim that every board or textbook uses the same sequence.

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