Solving and Graphing Quadratic Equations

What is the standard form of a quadratic function?

The standard form of a quadratic function is , where a, b, and c are real numbers, and . Each term in the function has a special purpose:

ax² is the quadratic term.

bx is the linear term.

c is the constant term.

The coefficient of the quadratic term, a, determines how wide or narrow the graphs are, and whether the graph turns upward or downward.

Important Tidbit

A positive quadratic coefficient causes the ends of the parabola to point upward.
A negative quadratic coefficient causes the ends of the parabola to point downward.
The greater the quadratic coefficient, the narrower the parabola.
The lesser the quadratic coefficient, the wider the parabola.

In this graph, coefficient a is positive and large. Therefore the parabola is narrow and points upward.

$Narrow parabola opening upward$

In this graph, coefficient a is smaller. Therefore, the parabola is wider.

$Wider parabola opening upward$

Here, coefficient a is negative, therefore the endpoints of the parabola point downward.

$Parabola opening downward$

The linear-term coefficient b shifts the axis of symmetry away from the y-axis. The direction of shift depends on the sign of the quadratic coefficient and the sign of the linear coefficient.

The axis of symmetry shifts to the right if the equation has:

positive a and negative b coefficients, or
negative a and positive b coefficients.

The axis of symmetry shifts to the left if the equation has:

positive a and positive b coefficients, or
negative a and negative b coefficients.

The constant term c affects the y-intercept. The greater the number, the higher the intercept point on the y -axis.

Question

Which graph best matches the quadratic function ?

A	$Parabola with linear coefficient -2 that opens upward$
B	$Parabola shifted left that opens upward$
C	$Parabola shifted right, opening down$
D	$Parabola shifted left, opening down$

Answer