Vertex Form and How to Graph a Parabola

Every quadratic graphs as a U-shaped curve called a parabola. Its turning point, the vertex, is the most useful point on the graph, and vertex form puts that point right in the equation.

Quick idea

In vertex form y = a(x − h)² + k, the vertex is (h, k). From standard form y = ax² + bx + c, the vertex sits at x = −b/(2a).

Three forms of a quadratic

Form	Equation	What it shows instantly
Standard	$y=ax^2+bx+c$	y-intercept $c$
Vertex	$y=a(x-h)^2+k$	vertex $(h,k)$
Factored	$y=a(x-r_1)(x-r_2)$	x-intercepts $r_1, r_2$

All three describe the same parabola; pick the form that answers the question you are asked.

Steps

1. Find the x-coordinate of the vertex with x = −b/(2a), or read h from vertex form.
2. Substitute that x into the equation to get the y-coordinate.
3. Check the sign of a: positive opens up, negative opens down.
4. Plot the vertex, the y-intercept, and any x-intercepts, then draw a smooth symmetric curve.

Worked example

Graph y = x² − 4x + 3 and write it in vertex form.

Full solution

Step-by-step walkthrough

Step 1

Here a = 1 and b = −4, so the vertex is at x = 4/2 = 2.

Step 2

Substitute x = 2: y = 4 − 8 + 3 = −1. The vertex is (2, −1).

Step 3

So y = (x − 2)² − 1. It opens up, crosses the y-axis at 3, and crosses the x-axis at x = 1 and x = 3.

More worked examples

Example 2: Graphing from vertex form

Sketch $y=-(x+1)^2+4$.

Step 1. Read the vertex. x + 1 means h = −1.

Step 2. a = −1, so it opens down and the vertex is a maximum.

Step 3. Find the x-intercepts by setting y = 0.

Step 4. Find the y-intercept by setting x = 0.

Answer: Plot (−1, 4), (1, 0), (−3, 0), and (0, 3), plus the mirror point (−2, 3), then draw a downward U through them.

Example 3: Writing the equation from a vertex and a point

A parabola has vertex (2, −3) and passes through (4, 5). Find its equation.

Step 1. Start from vertex form with h = 2 and k = −3.

Step 2. Substitute the point (4, 5) to find a.

Answer: $y=2(x-2)^2-3$

What a, h, and k do to the graph

• a controls direction and width. If $a>0$ the parabola opens up (the vertex is a minimum); if $a<0$ it opens down (the vertex is a maximum). If $|a|>1$ it is narrower than $y=x^2$; if $|a|<1$ it is wider.
• h shifts the graph left or right. $y=(x-2)^2$ moves right 2; $y=(x+2)^2$ moves left 2.
• k shifts the graph up or down.

The axis of symmetry is the vertical line $x=h$. Every point on one side has a mirror image on the other.

Converting standard form to vertex form

Two reliable routes:

1. Use the vertex formula. Find $h=-\frac{b}{2a}$, then $k$ by substituting $h$ into the equation. Keep the same $a$.
2. Complete the square. Factor $a$ out of the $x$ terms, add and subtract $(b/2)^2$ inside, and simplify.

For $y=2x^2-8x+5$: $h=\frac{8}{4}=2$, $k=2(4)-16+5=-3$, so $y=2(x-2)^2-3$.

Where the x-intercepts come from

The x-intercepts are the solutions of $ax^2+bx+c=0$, so you can find them by factoring or the quadratic formula. They sit symmetrically on either side of the vertex — their average is always $h$.

Common mistakes to avoid

• Watch the sign of h. In y = (x + 3)² − 2 the vertex is at x = −3, not 3, because x + 3 is the same as x − (−3).
• Plotting only the vertex. You need at least two more points (an intercept and its mirror image) to draw the right width.
• Forgetting that $a$ changes the width. $y=3(x-1)^2$ and $y=(x-1)^2$ share a vertex but are different parabolas.
• Drawing a V shape. A parabola is a smooth curve that is flat at the vertex.

Practice problems

1. Find the vertex of y = (x + 1)² + 4.
2. Find the vertex of y = x² + 6x + 5.
3. Does y = −2(x − 1)² + 3 open up or down, and what is its maximum value?

Answers

1. (−1, 4)
2. (−3, −4)
3. It opens down; the maximum value is 3, at x = 1.

Frequently asked questions

What is vertex form?

Vertex form is y = a(x − h)² + k. The vertex of the parabola is the point (h, k), and a tells you whether it opens up or down and how wide it is.

How do you find the vertex from standard form?

Use x = −b/(2a) for the x-coordinate, then substitute that value into the equation to get the y-coordinate.

What is the axis of symmetry?

It is the vertical line through the vertex, x = h (or x = −b/(2a) in standard form). The parabola is a mirror image on either side of it.

Ask PennPaper to explain it live

If the steps make sense but you still feel stuck, start a PennPaper lesson and ask the tutor to draw the problem on the whiteboard. Seeing the symbols move step by step is often what makes the concept click.

Related guides

• The Quadratic Formula Explained
• Completing the Square Explained Step by Step
• How to Graph a Linear Equation