Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 quadratic functions? Then this Algebra 1 lesson on Graphing Parabolas and Key Features explains each idea step by step, with worked examples.
Quick answer: A quadratic function graphs as a parabola. Its vertex is the lowest point when the leading coefficient is positive and the highest point when it is negative. The mirror line is x = −b2a. Standard form reveals the y-intercept, vertex form reveals the vertex, and factored form reveals the zeros.
Standards: F-IF.C.7a, F-IF.C.8a, F-IF.B.4, A-SSE.B.3b · Level: Grade 9 Algebra 1
What does the graph of a quadratic function look like?
When a ≠ 0, the rule f(x) = ax² + bx + c is quadratic. Its picture is a parabola, the U-shaped path of a basketball shot, or the area of a rectangle as one side stretches.
- Vertex: the turning point, a minimum when a > 0 and a maximum when a < 0.
- Axis of symmetry: a mirror line, x = h, that cuts the parabola into two matching halves.
- Intercepts: the points where the curve meets the x-axis or the y-axis.
Which three forms should you know?
- Standard form y = ax² + bx + c gives the point (0, c) at once, and the mirror line x = −b2a.
- Vertex form y = a(x − h)² + k shows the turning point (h, k).
- Factored form y = a(x − p)(x − q) shows crossings of the x-axis at p and q.
How do you find the vertex from standard form?
- Compute x = −b ÷ (2a), the x-coordinate of the turning point.
- Put that x back into the rule to get y.
- Plot (x, y), then add the intercepts to sketch the curve.
y = x² − 4x − 5. Here x = 4 ÷ 2 = 2 and y = 4 − 8 − 5 = −9, so the turning point is (2, −9). Factoring gives (x − 5)(x + 1), so the curve crosses the x-axis at 5 and −1 and the y-axis at −5.
How does completing the square give the vertex form?
Add and subtract the square of half the x-coefficient. For y = x² + 6x + 5, half of 6 is 3, so y = (x² + 6x + 9) − 9 + 5 = (x + 3)² − 4. The vertex is (−3, −4).
In (x + 3)² the vertex has x = −3, not 3. The sign inside the parentheses is the opposite of h.
The vertex sits halfway between the two zeros, so average the zeros to get the axis.
- The parabola opens up if a > 0 and down if a < 0.
- Axis of symmetry in standard form: x = −b2a.
- Vertex form y = a(x − h)² + k has vertex (h, k).
- Factored form y = a(x − p)(x − q) has zeros p and q.
- Completing the square turns standard form into vertex form.
Graphing Parabolas and Key Features: frequently asked questions
How do you find the vertex of a parabola?
In standard form, compute x = −b2a, then substitute it into the function to get y. In vertex form y = a(x − h)² + k, the vertex is simply (h, k).
What is the axis of symmetry?
It is the vertical line through the vertex that splits the parabola into two mirror-image halves. Its equation is x = h, where h is the x-coordinate of the vertex.
How do you know whether a parabola has a maximum or a minimum?
Look at the sign of a. If a is positive the graph opens up and the vertex is a minimum. If a is negative it opens down and the vertex is a maximum.
What does completing the square do?
It rewrites x² + bx + c as (x + b2)² + k, a perfect square plus a constant. This reveals the vertex and makes some equations easy to solve.
How can you find the zeros of a quadratic function?
Set the function equal to 0 and factor, if possible. In factored form y = a(x − p)(x − q) the zeros p and q can be read directly from the factors.
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