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Axis of Symmetry of a Parabola

Axis of Symmetry of a Parabola

The line passing through the focus and vertex of a parabola. The axis of symmetry is perpendicular to the directrix.

 

Example:  

Graph of upward-opening parabola with vertex at (3,-2), focus above vertex, vertical axis of symmetry at x=3, and horizontal...

This is a graph of the parabola Equation of a parabola: y = (1/4)x² − (3/2)x + 1/4 with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

Key Formula

x=−b2ax = -\frac{b}{2a}
Where:
  • aa = The coefficient of $x^2$ in the standard form $y = ax^2 + bx + c$
  • bb = The coefficient of $x$ in the standard form $y = ax^2 + bx + c$
  • xx = The $x$-value of every point on the axis of symmetry

Worked Example

Problem: Find the axis of symmetry of the parabola y=2x2−8x+5y = 2x^2 - 8x + 5.
Step 1: Identify the coefficients aa, bb, and cc from the equation y=ax2+bx+cy = ax^2 + bx + c.
a=2,b=−8,c=5a = 2, \quad b = -8, \quad c = 5
Step 2: Substitute aa and bb into the axis of symmetry formula.
x=−b2a=−−82(2)x = -\frac{b}{2a} = -\frac{-8}{2(2)}
Step 3: Simplify the expression.
x=−−84=84=2x = -\frac{-8}{4} = \frac{8}{4} = 2
Step 4: Write the equation of the axis of symmetry as a vertical line.
x=2x = 2
Answer: The axis of symmetry is the vertical line x=2x = 2.

Another Example

This example uses vertex form instead of standard form, showing that you can read the axis of symmetry directly from the vertex coordinates without using the −b/(2a)-b/(2a) formula.

Problem: Find the axis of symmetry of the parabola given in vertex form: y=−3(x+4)2+7y = -3(x + 4)^2 + 7.
Step 1: Recall that vertex form is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. The axis of symmetry passes through the vertex, so its equation is x=hx = h.
y=a(x−h)2+ky = a(x - h)^2 + k
Step 2: Rewrite the given equation to match vertex form. Note that (x+4)(x + 4) is the same as (x−(−4))(x - (-4)).
y=−3(x−(−4))2+7y = -3(x - (-4))^2 + 7
Step 3: Read off hh from the expression. Here h=−4h = -4.
h=−4h = -4
Step 4: Write the axis of symmetry.
x=−4x = -4
Answer: The axis of symmetry is the vertical line x=−4x = -4.

Frequently Asked Questions

How do you find the axis of symmetry from a graph?
Look for the vertex of the parabola — the highest or lowest point on the curve. The axis of symmetry is the vertical line that passes through that vertex. If the vertex is at (h,k)(h, k), the axis of symmetry is x=hx = h. You can also pick any two points on the parabola that share the same yy-value; the axis of symmetry runs through the midpoint of those two points.
Is the axis of symmetry always a vertical line?
Only when the parabola opens upward or downward (the standard case in most algebra courses). If a parabola opens left or right — written as x=ay2+by+cx = ay^2 + by + c — the axis of symmetry is a horizontal line of the form y=−b2ay = -\frac{b}{2a}. The key idea is that the axis of symmetry is always perpendicular to the directrix.
What is the difference between the axis of symmetry and the vertex?
The vertex is a point, specifically the turning point of the parabola, with coordinates (h,k)(h, k). The axis of symmetry is a line that passes through the vertex. For a vertical parabola, the vertex is (h,k)(h, k) and the axis of symmetry is the line x=hx = h. They are closely related but are different geometric objects — one is a point, the other is a line.

Axis of Symmetry vs. Directrix

Axis of SymmetryDirectrix
What it isA line that divides the parabola into two mirror-image halvesA fixed line used to define every point on the parabola
OrientationPerpendicular to the directrix (vertical for upward/downward parabolas)Perpendicular to the axis of symmetry (horizontal for upward/downward parabolas)
Passes through vertex?Yes — it always passes through the vertexNo — it lies on the opposite side of the vertex from the focus
Formula (vertical parabola)x=−b2ax = -\dfrac{b}{2a}y=k−14ay = k - \dfrac{1}{4a} (vertex form)

Why It Matters

You will use the axis of symmetry in nearly every problem involving quadratic functions, from graphing parabolas to finding maximum or minimum values. In physics, the axis of symmetry describes the trajectory midpoint of a projectile. Knowing this line also lets you quickly find the vertex and determine the optimal value in real-world optimization problems, such as maximizing profit or minimizing cost.

Common Mistakes

Mistake: Forgetting the negative sign in the formula and writing x=b2ax = \frac{b}{2a} instead of x=−b2ax = -\frac{b}{2a}.
Correction: The formula has a negative sign in front: x=−b2ax = -\frac{b}{2a}. A helpful check is to substitute your result back into the original equation — the axis should pass through the vertex.
Mistake: Confusing the sign of hh in vertex form. For example, reading y=(x+4)2y = (x + 4)^2 as h=4h = 4 instead of h=−4h = -4.
Correction: Vertex form is y=a(x−h)2+ky = a(x - h)^2 + k. Because of the subtraction, (x+4)(x + 4) means (x−(−4))(x - (-4)), so h=−4h = -4 and the axis of symmetry is x=−4x = -4.

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