What Changes do the coefficients have on the graph of quadratic equations?

Question

for algebra 2 use

Answer 1

Constant coefficient: shifts the graph up (for positive changes) or down (for negative changes)
Linear coefficient: shifts the graph left and up (for positive changes) or right and down (for negative changes).
Quadratic coefficient: Makes the graph thinner (when the absolute value of this increases) or fatter (when the absolute value of this decreases). Also, the sign determines whether the parabola points up (for positive values) or down (for negative values)

Answer 2

perhaps you meant quadratic functions and not equations?

quadratic functions take the form: y = Ax^2 + Bx + C,

while quadratic equations take the form: Ax^2 + Bx + C = 0.

the set of all ordered pairs {(x, y)} which satisfy the first functional equation comprises the graph of that quadratic function in the Cartesian plane.

to see the effects that changes to the coefficients {A, B, C} of the quadratic function have on its graph, rewrite the functional equation in standard form:

y = Ax^2 + Bx + C
y - C = A(x^2 + (B/A)x)
y - C = A(x^2 + (B/A)x + (B/2A)^2) - B^2/4A
y + (B^2 - 4AC/4A) = A(x + (B/A))^2

hence, the vertex of the graph of the quadratic function, which is a parabola, is at the point with coordinates: V(-B/A, (4AC - B^2)/4A). thus, we require that A <> 0 (i.e. "<>" means "is not equal to"), or else we would have a degenerate equation Bx + C = 0, which is an equation of the vertical line x = -(C/B) (where again we require that B <> 0).

since A <> 0, either A < 0 or A > 0. when A < 0, then the parabola opens downward, while if A > 0, the parabola opens upward. if B = 0, then the vertex of the parabola is at: (0, C).

also, if B^2 - 4AC > 0, then the parabola intersects the x-axis at two distinct points (i.e. at ( (-B - sqrt(B^2 - 4AC))/2A, 0 ) and (-B + sqrt(B^2 - 4AC))/2A, 0 )). if B^2 - 4AC = 0, then the parabola intersects the x-axis at exactly one point (i.e. at the point (-B/A, 0)). if B^2 - 4AC < 0, then the parabola does not intersect the x-axis. in particular, if B^2 - 4AC < 0 and A < 0, then the parabola lies entirely below the x-axis, while if B^2 - 4AC < 0 and A > 0, then the parabola lies entirely above the x-axis.

experimenting with a few illustrative quadratic functions would serve to reinforce understanding of the preceding concepts.

happy learning!

Answer 3

A quadratic equation will draw a parabola. If the coefficient of x^2 is positive, it will be a u shape. If the coefficient is negative, it will be upside down.

The larger the coefficient of x^2 is, the narrower the parabola will be. As the coefficient gets smaller, the parabola gets wider.

Answer 4

The coefficients can change the steepness of the curve as well as the roundness of the parabola. The higher the x^2 coefficient, the sharper the point. The higher the x coefficient, the steeper the sides.

Answer 5

In addition to the answers already given the coefficients can determine where and if the parabola crosses the x axis if it is in the form y=ax^2+bx=c.