A coordinate grid has been superimposed over the quadratic path of a basketball in
[link] . Find an equation for the path of the ball. Does the shooter make the basket?
(credit: modification of work by Dan Meyer)
The path passes through the origin and has vertex at
so
To make the shot,
would need to be about 4 but
he doesn’t make it.
Given a quadratic function in general form, find the vertex of the parabola.
Identify
Find
the
x -coordinate of the vertex, by substituting
and
into
Find
the
y -coordinate of the vertex, by evaluating
Finding the vertex of a quadratic function
Find the vertex of the quadratic function
Rewrite the quadratic in standard form (vertex form).
Rewriting into standard form, the stretch factor will be the same as the
in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the
from the general form.
The standard form of a quadratic function prior to writing the function then becomes the following:
Finding the domain and range of a quadratic function
Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all
y -values greater than or equal to the
y -coordinate at the turning point or less than or equal to the
y -coordinate at the turning point, depending on whether the parabola opens up or down.
Domain and range of a quadratic function
The domain of any
quadratic function is all real numbers unless the context of the function presents some restrictions.
The range of a quadratic function written in general form
with a positive
value is
or
the range of a quadratic function written in general form with a negative
value is
or
The range of a quadratic function written in standard form
with a positive
value is
the range of a quadratic function written in standard form with a negative
value is
Given a quadratic function, find the domain and range.
Identify the domain of any quadratic function as all real numbers.
Determine whether
is positive or negative. If
is positive, the parabola has a minimum. If
is negative, the parabola has a maximum.
Determine the maximum or minimum value of the parabola,
If the parabola has a minimum, the range is given by
or
If the parabola has a maximum, the range is given by
or
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The term pathogen came into use in the 1880s.[1][2
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