a\cdot x^2 + b\cdot x + c
is graphed below. Determine the signs of a
, b
, and c
.
a
is (A > 0) ? 'positive' : 'negative'
b
is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' )
c
is (C > 0) ? 'positive' : ( (C < 0) ? 'negative' : 'zero' )
The number a
determines how the legs are oriented. Is the parabola smiling or frowning?
A smiling parabola means a
is positive and a frowning parabola means a
is negative.
The parabola is ( A > 0) ? "smiling" : "frowning" thus a
is ( A > 0 ) ? "positive" : "negative".
The number c
determines where the parabola intersects the y
-axis. Is the positive or negative part?
If the parabola intersected the positive part of the y
-axis, then c
would be positive.
The parabola intersects the y
-axis in the point (0,c) = (0,C)
, thus c
is ( C > 0 ) ? 'positive' : ( (C<0) ? 'negative' : 'zero' ).
The number b
determines how the parabola intersects the y
-axis. Imagine the tangent at the intersection. What is the slope?
The tangent where the parabola intersects the y
-axis was drawn in orange. The number b
is the slope.
The tangent has a (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ) slope, so b
is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ).
a
is ( A > 0 ) ? "positive" : "negative", b
is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ), and c
is (C > 0) ? 'positive' : ( (C < 0) ? 'negative' : 'zero' ).