17.3 What is ‘b’ in Quadratic Equaton

17.3 What is ‘b’ in Quadratic Equaton?

In a quadratic equation,$$y=ax^2+bx+c$$

the b is the coefficient of xx.

For example, in:$$y=x^2-4x+3$$

we have:$$a=1,\qquad b=-4,\qquad c=3$$

So here:$$b=-4$$

🟦 What does $b$ do?

The value of $b$ helps determine the horizontal position of the turning point and the line of symmetry.

The x-coordinate of the turning point is:$$x=-\frac{b}{2a}$$

For:$$y=x^2-4x+3$$$$a=1,\qquad b=-4$$

Therefore:$$x=-\frac{-4}{2(1)}$$$$x=2$$

So the line of symmetry is:$$x=2$$

and the turning point also has x-coordinate $2$.

🟩 Another Example

Consider:$$y=2x^2+8x+5$$

Here:$$a=2$$$$b=8$$$$c=5$$

Find the line of symmetry:$$x=-\frac{8}{2(2)}$$$$x=-2$$

So the turning point lies on:$$x=-2$$

⭐ Easy Way to Remember

For:$$y=ax^2+bx+c$$

• $a$ controls whether the parabola opens up or down and how wide it is.
• $b$ helps determine the horizontal position of the turning point.
• $c$ gives the y-intercept.

Quick Example

$$y=3x^2-6x+7$$

Therefore:$$a=3,\qquad b=-6,\qquad c=7$$

So bb is simply the number multiplying xx.

One important point: if there is no $x$-term, then:$$b=0$$

For example:$$y=x^2+4$$

has: b=0