Standard form
The equation for standard form is written as y=ax2+bx+c
An equation written in standard form is yet another equation that forms a parabola when graphed. Each letter in the standard form equation tells us a piece of information about the parabola, just like the letters from the vertex form equation had.
An equation written in standard form is yet another equation that forms a parabola when graphed. Each letter in the standard form equation tells us a piece of information about the parabola, just like the letters from the vertex form equation had.
- In standard form the “a” value
represents the same thing as it does in vertex form. Just as mentioned before,
it dictates the direction of the parabola’s opening.
If a > 0 the parabola opens upwards, resulting in a minimum value. If a <
0 on the other hand, the parabola opens downwards, resulting in a maximum
value. The “a” value can never equal 0, because that would result in the
formation of a straight line, not a
parabola. “a”, can also tell us the
width of a parabola. The smaller the value of “a”, the wider the parabola
becomes (fractions would result in a wider parabola due to this), but as the
value of “a” increases, the parabola becomes narrower.
- The “b” value translates the parabola horizontally
across the x-axis. If the “b” value is positive, the parabola moves to the
left, and if it’s negative the “b” value moves to the right. In the picture seen on the left the “b” value
is positive (y=2x2+5x+1), causing the parabola to move
left.
- The “c” value represents the y-intercept, where the
parabola crosses the y-axis. This picture supports this, the “c” value in the
equation is 3, and that’s exactly where the parabola intercepts with the y-axis.
- The “x” and “y” values in this equation represent
the coordinates of the vertex. “x” represents the unknown variable which can
easily be found using the equation x = -b/2a, from there the x value can be
substituted into the equation in order to find “y”. A demonstration of this
process is shown in the picture.
Vertex form may be great
for finding information about your parabola quickly and easily, but standard
form proves to be useful when you need to find the x-intercepts of a parabola,
simply because an equation in this form is easily factored. The YouTube video below shows us how to do
this. The solutions that Nancy finds in the end, after factoring the equations, are our x-intercepts.