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Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.

The standard form of a quadratic function presents the function in the form

f ( x ) = a ( x h ) 2 + k

where ( h ,   k ) is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function    .

As with the general form, if a > 0 , the parabola opens upward and the vertex is a minimum. If a < 0 , the parabola opens downward, and the vertex is a maximum. [link] represents the graph of the quadratic function written in standard form as y = −3 ( x + 2 ) 2 + 4. Since x h = x + 2 in this example, h = –2. In this form, a = −3 , h = −2 , and k = 4. Because a < 0 , the parabola opens downward. The vertex is at ( 2 ,  4 ) .

Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.

The standard form is useful for determining how the graph is transformed from the graph of y = x 2 . [link] is the graph of this basic function.

Graph of y=x^2.

If k > 0 , the graph shifts upward, whereas if k < 0 , the graph shifts downward. In [link] , k > 0 , so the graph is shifted 4 units upward. If h > 0 , the graph shifts toward the right and if h < 0 , the graph shifts to the left. In [link] , h < 0 , so the graph is shifted 2 units to the left. The magnitude of a indicates the stretch of the graph. If | a | > 1 , the point associated with a particular x - value shifts farther from the x- axis, so the graph appears to become narrower, and there is a vertical stretch. But if | a | < 1 , the point associated with a particular x - value shifts closer to the x- axis, so the graph appears to become wider, but in fact there is a vertical compression. In [link] , | a | > 1 , so the graph becomes narrower.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.

a ( x h ) 2 + k = a x 2 + b x + c a x 2 2 a h x + ( a h 2 + k ) = a x 2 + b x + c

For the linear terms to be equal, the coefficients must be equal.

–2 a h = b ,  so  h = b 2 a .

This is the axis of symmetry    we defined earlier. Setting the constant terms equal:

a h 2 + k = c            k = c a h 2              = c a ( b 2 a ) 2              = c b 2 4 a

In practice, though, it is usually easier to remember that k is the output value of the function when the input is h , so f ( h ) = k .

Forms of quadratic functions

A quadratic function is a function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function    is f ( x ) = a x 2 + b x + c where a , b , and c are real numbers and a 0.

The standard form of a quadratic function    is f ( x ) = a ( x h ) 2 + k .

The vertex ( h , k ) is located at

h = b 2 a ,   k = f ( h ) = f ( b 2 a ) .

Given a graph of a quadratic function, write the equation of the function in general form.

  1. Identify the horizontal shift of the parabola; this value is h . Identify the vertical shift of the parabola; this value is k .
  2. Substitute the values of the horizontal and vertical shift for h and k . in the function f ( x ) = a ( x h ) 2 + k .
  3. Substitute the values of any point, other than the vertex, on the graph of the parabola for x and f ( x ) .
  4. Solve for the stretch factor, | a | .
  5. If the parabola opens up, a > 0. If the parabola opens down, a < 0 since this means the graph was reflected about the x - axis.
  6. Expand and simplify to write in general form.
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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