How to Graph a Parabola: All Three Forms
Every quadratic graphs as a parabola. Which form the equation is written in decides which features you can read off immediately, and picking the right one is most of the work. This guide covers all three forms, the fastest route to the vertex, and the three mistakes that account for most lost marks.
What a parabola looks like and why
A quadratic is any equation where the highest power of x is 2.
Graphed, it always produces the same symmetric U shape, called a parabola.
The curve has a single turning point, the vertex, and it is
a perfect mirror image about the vertical line through that vertex.
That symmetry is not decoration; it is the most useful property a parabola has. Find one point on the curve and you get a second one free, reflected across the axis of symmetry.
The coefficient of x² controls two things at once. Its sign
decides which way the curve opens, and its size decides how narrow the curve
is. A coefficient of 3 gives a parabola three times as steep as
y = x²; a coefficient of 0.2 gives a wide, shallow one.
Standard form: y = ax² + bx + c
This is the form you will most often be handed. Two things can be read from it without any work:
- The direction. Positive
aopens upward, negativeaopens downward. - The y-intercept. It is
c, always. Settingx = 0kills the other two terms.
The vertex takes one short calculation:
x-coordinate of the vertex:
x = -b / (2a)
Then substitute that x back into the original equation to get
the y-coordinate. The vertical line x = -b/(2a) is
the axis of symmetry.
Worked example
Graph y = 2x² - 8x + 5.
- Direction:
a = 2, which is positive, so the parabola opens upward and the vertex is a minimum. - Axis of symmetry:
x = -(-8) / (2 × 2) = 8 / 4 = 2. - Vertex y-value:
y = 2(2²) - 8(2) + 5 = 8 - 16 + 5 = -3. The vertex is(2, -3). - y-intercept:
c = 5, so the curve passes through(0, 5). - Mirror it:
(0, 5)is two units left of the axis, so(4, 5)is on the curve too.
Vertex form: y = a(x - h)² + k
Vertex form hands you the vertex directly: it is (h, k). No
calculation at all.
The one thing that trips people up is the sign. The form has a
minus inside the bracket, so y = (x - 3)² has its
vertex at x = +3, and y = (x + 3)² has its vertex
at x = -3. The graph moves the opposite way to the sign you
see.
| Equation | Vertex | What changed |
|---|---|---|
y = x² | (0, 0) | The starting point |
y = (x - 3)² | (3, 0) | Three units right |
y = (x - 3)² + 2 | (3, 2) | Three right, two up |
y = -2(x - 3)² + 2 | (3, 2) | Same vertex, now opening downward and twice as steep |
See it move. Open the calculator and type
a(x - h)^2 + k. It will offer you a slider for each of
a, h and k. Dragging them one at a
time is the quickest way to make vertex form stick.
Try it now.
Factored form: y = a(x - p)(x - q)
Factored form hands you the x-intercepts: they are
p and q. The curve crosses the x-axis at exactly
those two values, because that is where one of the brackets becomes zero.
And because of the symmetry, the vertex sits exactly halfway between them:
xof the vertex= (p + q) / 2
So y = (x - 1)(x + 5) crosses at x = 1 and
x = -5, and its vertex is at x = (1 + (-5))/2 = -2.
Substituting gives y = (-3)(3) = -9, so the vertex is
(-2, -9).
Graphing one by hand in four steps
This works for any quadratic in any form, and gives a sketch good enough for full marks.
- Decide the direction. Look at the sign of the squared term. Up or down.
- Find the vertex using whichever route the form gives you:
read it off vertex form, use
-b/(2a)from standard form, or average the roots from factored form. - Plot the y-intercept by setting
x = 0, then reflect it across the axis of symmetry for a free second point. - Draw a smooth curve through the three points. Parabolas have no straight sections and no corners.
Three mistakes that cost marks
1. Getting the shift backwards
y = (x - 4)² moves the curve right by 4, not left.
Everything inside the bracket does the opposite of what it looks like.
Checking one point settles it every time: at x = 4 the bracket
is zero, which is the lowest the squared term can be, so the vertex must be
there.
2. Writing -x² when you mean (-x)²
-x² means -(x²), so at x = 3 it is
-9. (-x)² means the square of -x, which is +9. The
first opens downward, the second is identical to y = x². Our
calculator follows the standard convention, so typing -x^2
gives you the downward parabola.
3. Drawing a curve with a pointed bottom
A parabola is smooth at its vertex. A sharp corner is the graph of
y = |x|, which is a different function entirely. If your sketch
has a point at the bottom, you have drawn an absolute value graph.
Compare the two here.
Questions
How do you find the vertex of a parabola from standard form?
Use x = -b/(2a) to get the x-coordinate, then substitute that back into the equation to get the y-coordinate. For y = 2x^2 - 8x + 5, x = 8/4 = 2 and y = 2(4) - 16 + 5 = -3, so the vertex is (2, -3).
Which way does a parabola open?
Upward when the coefficient of x squared is positive, downward when it is negative. The sign of a is the only thing that decides it.
Can a parabola have no x-intercepts?
Yes. If the discriminant b squared minus 4ac is negative, the curve never crosses the x-axis, and the quadratic has no real roots. It still has exactly one y-intercept.
What is the axis of symmetry?
The vertical line through the vertex, written x = -b/(2a). The parabola is a mirror image on either side of it, which is why the two x-intercepts are always the same distance from it.
How do I graph a parabola quickly by hand?
Find the vertex, draw the axis of symmetry, plot the y-intercept, then mirror that point across the axis. Three points and the symmetry are enough for a good sketch.