Relating Equations and Graphs for Quadratic Functions
Quadratic Equations
A quadratic equation is an equation that can be simplified to follow the pattern:
y = a · x 2 + b
· x + c,
where the letter x represents the input, the letter y represents the value of the output and
the letters a, b and c are all numbers. Sometimes the numbers a, b and c are referred to as
coefficients.
For an equation to be a quadratic equation, the number that multiplies the x 2, that is, the
number a cannot be equal to zero. It is okay for either or both of the numbers b and c to
be equal to zero, but a cannot be equal to zero.
Sometimes you might see a quadratic equation written in a format like:
y = x 2 + 2 · x + 3,
and wonder if this is okay, as there does not seem to be a value of a multiplying in front
of the x 2. In this situation everything is okay, because the value of the number a in this
situation is a = 1. We could write 1 · x 2 in the quadratic equation, but because a factor of
“1†does not change the value of anything, the factor of “1†in 1
· x 2 is usually just left out.
Other Ways of Writing Quadratic Equations
The format for a quadratic equation given above,
y = a · x 2 + b
· x + c,
where the letter x represents the input, the letter y represents the value of the output and
the letters a, b and c are all numbers, is called standard form.
Other ways of writing the equations for quadratic functions include vertex form,
y = a · (x - h) 2 + k,
where the letter x represents the value of the input, the letter y represents the value of the
output and the letters a, h and k all represent numbers. Just as in standard form, in vertex
form the number a cannot be equal to zero. Converting a quadratic equation to vertex
form is often quite helpful as it allows you to determine exactly where the graph of the
quadratic equation reaches its “low point†or “high point†very easily. Every single
quadratic formula can be converted to vertex form. The process for doing this conversion
is called completing the square.
The third common way of writing the formula for a quadratic function is called
factored
form in which the quadratic function is written as a product of two factors,
y = a · (x - c)
· (x - d).
Many (but not all) quadratic functions can be written in factored form. The quadratic
functions that can be written in factored form are the ones whose graphs touch the x-axis
in at least one place. If the graph of the quadratic function never touches the x-axis then
the formula of the quadratic function cannot be converted to factored form.
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