Vertex form is a way to write a quadratic function as y = a(x − h)2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)2 + 1 has its vertex at (3, 1).
In Spanish
forma de vértice
Also called
vertex form of a quadratic, vertex form of a parabola, standard form a(x – h)^2 + k, graphing form
In y = (x − 2)2 − 3, the −2 inside the square moves the parent graph right 2 and the −3 outside moves it down 3, so the vertex is (2, −3).
Formal definition and notation
A quadratic function is in vertex form when it is written f(x) = a(x − h)2 + k with a ≠ 0. Its graph is the graph of y = ax2 translated h units horizontally and k units vertically, so the vertex is (h, k) and the axis of symmetry is x = h. If a > 0, k is the minimum value; if a < 0, k is the maximum value. Any quadratic ax2 + bx + c can be rewritten this way by completing the square, with h = −b2a.
the x-coordinate of the vertex: the horizontal shift
k
the y-coordinate of the vertex: the vertical shift and the max or min value
Read h with the opposite sign of what you see: (x − 4)2 means h = 4, and (x + 4)2 = (x − (−4))2 means h = −4. The sign of k is read as written.
Try it: run y = -(x – 2)^2 + 4
Interactive
The machine computes y = −(x − 2)2 + 4 in vertex form order: subtract 2, square, multiply by −1, add 4. It has run 0, 1, 3 and 4. Try 2: it gives the largest output, 4, because (2, 4) is the vertex. Notice that 1 and 3 give the same output.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Vertex Form examples, step by step
Example 1: Read the vertex
Warm-up
Find the vertex of y = 3(x + 1)2 + 5. Is it a maximum or a minimum?
Rewrite the inside as x − h: x + 1 = x − (−1), so h = −1.
The constant outside is k = 5.
a = 3 > 0, so the parabola opens up and the vertex is a minimum.
Answer: The vertex is (-1, 5), a minimum: the smallest value of y is 5.
Example 2: Complete the square
Core
Write y = x2 + 8x + 11 in vertex form and give the vertex.
Take half of the x-coefficient and square it: ( 82 )2 = 16.
Add and subtract 16: y = (x2 + 8x + 16) − 16 + 11.
Half the x-coefficient only works after the leading coefficient is factored out. Here the vertex is (−3, −15).
Not to be confused with
Vertex Formvs.Standard form of a quadratic
Standard form y = ax2 + bx + c shows the y-interceptc. Vertex form y = a(x − h)2 + k shows the vertex. y = x2 − 4x + 7 and y = (x − 2)2 + 3 are the same function.
Vertex Formvs.Factored form
Factored form y = a(x − r)(x − s) shows the x-intercepts r and s. Vertex form shows the turning point. y = (x − 1)(x − 5) equals y = (x − 3)2 − 4.
The vertex is a point, such as (2, −3). Vertex form is a way of writing the whole equation so that point is visible.
Quick check: 3 questions
Question 1 of 3
What is the vertex of y = −2(x − 5)2 + 7?
Show answer and explanation
Answer: C.x − 5 gives h = 5 and the constant gives k = 7. (−5, 7) flips the sign of h, (5, −7) flips k, and −2 is a, not a coordinate.
Question 2 of 3
Which is y = x2 − 6x + 2 in vertex form?
Show answer and explanation
Answer: B. Half of −6 is −3 and (−3)2 = 9: (x − 3)2 − 9 + 2 = (x − 3)2 − 7. The first choice forgets to subtract 9, the third has the wrong sign, and the last uses all of −6 instead of half.
Question 3 of 3
What is the maximum value of y = −12(x + 1)2 + 9?
Show answer and explanation
Answer: D. Since a < 0, the parabola opens down, and its highest value is k = 9, reached at x = −1. −1 is where the maximum happens, not the maximum itself.
Vertex Form: frequently asked questions
How do you convert standard form to vertex form?
Complete the square. For y = x2 + 10x + 4, take half of 10, which is 5, and square it to get 25. Add and subtract 25: y = (x2 + 10x + 25) − 25 + 4 = (x + 5)2 − 21. A faster check: h = −b2a = −5 and k = f(−5) = −21.
Why is it called vertex form?
Because the vertex of the parabola appears right in the equation. In y = a(x − h)2 + k, the squared part is never negative, so it is smallest, 0, exactly when x = h. At that input y = k. That turning point (h, k) is the vertex, and you can read it without any calculation.
How do you find the vertex from vertex form when there is a plus sign inside?
Rewrite the plus as minus a negative. y = (x + 7)2 − 2 is y = (x − (−7))2 − 2, so h = −7 and the vertex is (−7, −2). The rule of thumb: the x-coordinate is the number that makes the inside of the parentheses equal 0.
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