randRangeNonZero( -5, 5 ) randRange( -5, 5 ) randRange( -5, 5 ) -2 * A * H A * H * H + K
( function( x ){ return B * x + C; }) "initialized later" [$._("positive"), $._("zero"), $._("negative")] (A > 0) ? CHOICES[0] : CHOICES[2] (B > 0) ? CHOICES[0] : ( (B < 0) ? CHOICES[2] : CHOICES[1] ) (C > 0) ? CHOICES[0] : ( (C < 0) ? CHOICES[2] : CHOICES[1] )

a\cdot x^2 + b\cdot x + c is graphed below. Determine the signs of a, b, and c.

graphInit({ range: [ [ -10, 10 ], [ -10, 10 ] ], scale: [ 25, 25 ], ticks: false, labels: false }); // graph style({ stroke: "#FF0000", strokeWidth: 2}); plot( function(x) { return A * x * x + B * x + C; }, [ -10,10 ] );

a is A_SOLN

b is B_SOLN

c is C_SOLN

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 smiling thus a is positive.

The parabola is frowning thus a is 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_SOLN.

The number b determines how the parabola intersects the y-axis. Imagine the tangent at the intersection. What is the slope?

style({ stroke: "#FF8800", strokeWidth: 2}); line( [ -10, F( -10 )], [ 10, F( 10 )]);

The tangent where the parabola intersects the y-axis was drawn in orange. The number b is the slope.

The tangent has a B_SOLN slope, so b is B_SOLN.

a is A_SOLN, b is B_SOLN, and c is C_SOLN.