How to Graph a Parabola

Every quadratic y = ax² + bx + c draws the same basic shape: a smooth U with one turning point. What changes is whether it opens up or down, how narrow it is, and where it sits on the grid. All of that is determined by three numbers.

Step 1 — Direction and width from a

The coefficient a decides two things before you calculate anything else:

• Sign — positive opens upward, negative opens downward.
• Magnitude — the larger |a|, the narrower the curve. A parabola with a = 3 is three times steeper than one with a = 1.

If the equation is in vertex form y = a(x − h)² + k, you can read the vertex straight off — no calculation needed.

Step 2 — Find the vertex

The vertex is the turning point: the minimum if the parabola opens up, the maximum if it opens down.

From vertex form

In y = a(x − h)² + k the vertex is (h, k) directly. Watch the sign — for y = (x − 3)² + 1 the vertex is (3, 1), not (−3, 1). The minus is already inside the bracket.

From standard form

In y = ax² + bx + c, the x-coordinate of the vertex is −b / 2a. Substitute that value back into the equation to get the y-coordinate.

Step 3 — Axis of symmetry

The axis of symmetry is the vertical line through the vertex, x = −b / 2a. The parabola mirrors perfectly across it, which is why finding one root immediately gives you the other.

Step 4 — Where it crosses the x-axis

The roots are the points where y = 0. Solve ax² + bx + c = 0 with the quadratic formula, and let the discriminant b² − 4ac tell you what to expect:

• Positive — two distinct real roots. The parabola crosses the x-axis twice.
• Zero — one repeated root. The vertex sits exactly on the x-axis and the curve just touches it.
• Negative — no real roots. The curve passes entirely above or below the axis.

Step 5 — The y-intercept

Set x = 0 and the equation collapses to y = c. That is the y-intercept, and it is always the constant term in standard form. It is the quickest single point you can find.

Putting it together

With the direction, vertex, axis and intercepts in hand you can sketch a parabola accurately without plotting a single intermediate point. Plot the vertex, mark the y-intercept, reflect it across the axis to get a second point, and draw the U through them.

To check your sketch against the real curve, open the graphing calculator and enter the equation — the vertex and roots should match what you worked out.

Common mistakes

Dropping the sign in vertex form

(x − h)² means the vertex x-coordinate is h, not −h. The negative is already part of the form.

Forgetting the 2a in the denominator

The vertex x-coordinate is −b / 2a, not −b / a. This is the most common arithmetic slip in the whole topic.

Reading width off the wrong coefficient

Only a controls the width. Neither b nor c changes how narrow the parabola is — they only move it around.