How do you find the vertex and the intercepts of a parabola?

The link between equation and graph

Solving a quadratic equal to zero is asking where the function takes the value zero — that is, where the graph crosses the horizontal axis. The same question in two languages.

It follows that the discriminant describes the graph too. Two solutions mean two crossings, one solution means the parabola touches the axis, and none means it never reaches it.

The vertical intercept is simpler still. Substitute zero for x and every term containing x vanishes, leaving the constant term on its own.

The vertex and the axis of symmetry

A parabola is symmetric, so the vertex sits exactly midway between the two crossings. Its x-coordinate is minus the middle coefficient divided by twice the leading coefficient.

Once you have the x-coordinate, substitute it into the function to get the y-coordinate. The vertical line through the vertex is the axis of symmetry.

Which way it opens

A positive leading coefficient means the parabola opens upwards and the vertex is its lowest point. A negative one turns the picture over, and the vertex becomes the highest point.

That also answers the question of where the function is positive. An upward parabola is positive outside the interval between its two crossings; a downward one is positive between them.

Worked examples

1. Find where the parabola x squared minus 4 meets both axes

   1. For the horizontal axis: solve x squared minus 4 equals zero
   2. Factorise as a difference of squares: (x + 2)(x − 2)
   3. For the vertical axis: substitute zero, giving minus 4

   Answer: It meets the horizontal axis at 2 and at minus 2, and the vertical at minus 4

2. Find the vertex of the parabola x squared minus 6x plus 5

   1. Coefficients: 1 and minus 6
   2. x-coordinate: minus, minus 6, over twice 1, which is 3
   3. Substitute 3: 9 minus 18 plus 5

   Answer: The vertex is (3, −4), and the axis of symmetry is x equals 3

3. Does the parabola minus x squared plus 9 open up or down, and where is it positive?

   1. The leading coefficient is minus 1, which is negative, so it opens downwards
   2. Crossings with the horizontal axis: 3 and minus 3
   3. A downward parabola is positive between its crossings

   Answer: It opens downwards and is positive for x between minus 3 and 3

Common mistakes

Forgetting the minus in the vertex formula

The x-coordinate is minus the middle coefficient over twice the leading one. Dropping the minus reflects the vertex across the vertical axis, and the whole graph lands in the wrong place.

Stopping after the x-coordinate of the vertex

A vertex is a point and so has two coordinates. The second comes from substituting into the function, and without it you can neither sketch the graph nor answer a maximum or minimum question.

Assuming every parabola crosses the horizontal axis

A negative discriminant means no crossing at all. The parabola then lies entirely above the axis or entirely below it, depending on which way it opens.

More in Quadratic Equations

• How do you expand brackets and what are the special products?
• How do you factorise an algebraic expression?
• How do you solve a quadratic equation by factorising?
• What is the quadratic formula and how do you use it?
• What is the discriminant and how many solutions does an equation have?