Graphing Parabolas Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Graphing Parabolas.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The process of plotting a quadratic function by identifying its key features: vertex, axis of symmetry, direction of opening, y-intercept, and x-intercepts (if they exist).

A parabola is a U-shaped curve (or upside-down U). Start by finding the vertex—that is the turning point. Then the axis of symmetry tells you the curve is a mirror image on both sides. Plot a few symmetric points and connect them in a smooth curve.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A parabola is completely determined by a few key features. Finding those features makes graphing systematic rather than point-by-point.

Common stuck point: Determining whether the parabola opens up or down and finding the vertex from standard form (use x = -\frac{b}{2a}).

Sense of Study hint: Plot the vertex first, then the y-intercept, then use symmetry to find the mirror-image point.

Worked Examples

Example 1

easy
Identify the key features of f(x) = x^2 - 4x + 3 for graphing.

Solution

  1. 1
    Direction: a = 1 > 0, opens upward.
  2. 2
    Vertex: x = -\frac{-4}{2} = 2; f(2) = 4 - 8 + 3 = -1. Vertex: (2, -1).
  3. 3
    y-intercept: f(0) = 3, point (0, 3).
  4. 4
    x-intercepts: factor x^2 - 4x + 3 = (x-1)(x-3) = 0, so x = 1 and x = 3.

Answer

Vertex (2,-1), opens up, x-intercepts at 1 and 3, y-intercept at 3.
To graph a parabola, find: (1) direction from the sign of a, (2) vertex, (3) y-intercept at x=0, (4) x-intercepts by setting f(x) = 0.

Example 2

medium
Sketch g(x) = -x^2 + 2x + 3. Find vertex and intercepts.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Does f(x) = 5x^2 + 1 have any x-intercepts?

Example 2

medium
The vertex of a parabola is (0, -4) and it passes through (2, 0). Find the equation.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic vertex formcoordinate plane