Graphing Quadratic Functions

Q: How do I find the axis of symmetry?

Use $x = -\dfrac{b}{2a}$ with $a$ and $b$ from standard form. It is the vertical line through the vertex.

Q: How do I get the vertex's y-value?

Substitute the axis-of-symmetry x-value back into the function. The result is the vertex's y-coordinate.

Q: Which way does the parabola open?

Up if $a > 0$ (vertex is a minimum), down if $a < 0$ (vertex is a maximum).

A quadratic in standard form $f(x) = ax^2 + bx + c$ graphs as a parabola. Three landmarks pin it down: the axis of symmetry, the vertex on that axis, and the $y$ -intercept.

The axis of symmetry is $x = -\dfrac{b}{2a}$ . Everything else follows from it, so that formula is the anchor of this lesson.

Axis of symmetry and vertex

The axis of symmetry is the vertical line $x = -\dfrac{b}{2a}$ . The parabola is a mirror image across it.

The vertex sits on that axis. Find its x-value with the formula, then plug back into the function to get the y-value — that is the highest or lowest point.

Below, the parabola $y = x^2 - 2x - 3$ is mirrored across its axis of symmetry $x = 1$ (dashed), with the vertex $(1, -4)$ resting on that line.

Direction and intercepts

If $a > 0$ the parabola opens up (the vertex is a minimum); if $a < 0$ it opens down (a maximum). The $y$ -intercept is simply $c$ .

With the vertex, the direction, and the $y$ -intercept, plus symmetry, you can sketch the whole parabola quickly.

Worked examples

Example 1: axis of symmetry

Find the axis of symmetry of $f(x) = x^2 - 6x + 5$ .

Use x = -b/(2a)

x = -\dfrac{-6}{2(1)}

Simplify

x = 3

Answer: $x = 3$

Example 2: the vertex

Find the vertex of $f(x) = x^2 - 6x + 5$ .

Axis of symmetry gives x

x = 3

Plug in for y

f(3) = 9 - 18 + 5 = -4

Answer: $(3, -4)$

Example 3: a parabola that opens down

Which way does $f(x) = -2x^2 + 8x - 5$ open, and what is its $y$ -intercept?

Check the sign of a

a = -2 < 0 \text{, opens down}

The y-intercept is c

(0, -5)

Answer: It opens down, so the vertex is a maximum; the $y$ -intercept is $(0, -5)$

Try one yourself

Common questions

How do I find the axis of symmetry?

Use $x = -\dfrac{b}{2a}$ with $a$ and $b$ from standard form. It is the vertical line through the vertex.

How do I get the vertex's y-value?

Substitute the axis-of-symmetry x-value back into the function. The result is the vertex's y-coordinate.

Which way does the parabola open?

Up if $a > 0$ (vertex is a minimum), down if $a < 0$ (vertex is a maximum).

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Graphing Quadratic Functions

Axis of symmetry and vertex

Direction and intercepts

Worked examples

Example 1: axis of symmetry

Example 2: the vertex

Example 3: a parabola that opens down

Try one yourself

Common questions

Keep learning

Completing the Square

Solving Quadratic Equations by Graphing

Quadratic Inequalities