Quadratic Equations?

Question

f(x) = -2x^2 - x +6

Find the equation of the axis of symmetry.

Answer 1

Can this be factored?

(-2x + 3)(x + 2)

This is a vertical parabola with points at (3/2, 0) and (-2, 0). The (vertical) axis of symmetry is halfway between, at

(3/2 - 2)/2, or -1/4.

We can check by taking the first derivative,

-4x - 1 = 0

and you see that the vertex has x = -1/4.

The axis of symmetry is x = -1/4.

Answer 2

f(x) = -2x^2 - x + 6

This is a parabola with a downward concavity because the sign of x^2 is negative. Thus it has a maximum point.

The axis of symmetry is a vertical line that passes through the x-value of the maximum point (or minimum point if that was the case).

To find this maximum point, you have to differentiate the function to get the slope equation. Setting this slope equation to zero (because the slope is zero at a maximum or minimum point), will then give you the x-value.

So, differentiating gives :
-4x - 1

Setting equal to zero gives :
-4x - 1 = 0

Therefore, x = -1 / 4. This is the equation of the axis of symmetry.

Or, if you're not yet into calculus, the function can be factored as :
f(x) = -2x^2 - x + 6 = (x + 2)(-2x + 3)

Setting this equal to 0 gives the 2 zeros at x = -2 and x = 3 / 2.

The maximum point (or minimum) is always halfway between the x-values of the two zeros.

Halfway between = their sum, divided by 2
= (-2 + 3 / 2) / 2
= - 1 / 4

That is, at x = - 1 / 4.

Answer 3

Use Complete the square method:
**Make sure the coefficient of x² is 1.
-2x² - x + 6
= -2[x² + ½x -3]
= -2[x² + ½x + (¼)² -3 - (¼)²]
**The coefficient of x is divided by 2 and the result is squared. This squared result is added and subtracted in the equation so the equation remains the same.
= -2[(x + ¼)² - 3⅛]
= -2(x + ¼)² + 6¼

Using this method, the MAX point of the curve will be the point when -2(x + ¼)² = 0. Which will be the point where x = -¼

Hope this helps=)

Alternatively, you can factorize it, and find the coordinates where the curve cuts the y axis. f(x)=(-2x + 3)(x + 2)
Afterwhich you find the mid point of this 2 coordinates.

Hope this helps as well=)

Answer 4

Axis of symmetry passes through the point where the derivative of the parabola (i.e. -4x-1 ) is zero, which is x=-1/4. Looking at the equation you know it is symmetric about a line parallel to y axis. So the equation of the axis of symmetry is the line x=-1/4.

Answer 5

f(x)=-2x^2-x+6
=-2(x^2+1/2x+1/4-1/4)+6
y=-2(x+1/2)^2+13/2
setting y=0
0=-2x^2-4x+3x+6
0=-2x(x+2)+3(x+2)
=(x+2)(-2x+3)=0
x=-2 or 3/2
the axis of symmetry=x=-1/4