Question

The vertex of a parabola is (-1.5, -12.5), and its y-intercept is (0, -8).

Answer 1

Answer:

The x-intercept are x=-4,1.                    

Step-by-step explanation:

Given : The vertex of a parabola is (-1.5, -12.5), and its y-intercept is (0, -8).

To find : The x-intercepts of the parabola are ?

Solution :

The general form of the parabola equation is  

$y=a(x-h)^2+k$ ......(1)

where, a is the initial value and (h,k) are vertex of the parabola.

We have given the vertex (h,k)=(-1.5, -12.5)

Substitute in equation (1),

$y=a(x-(-1.5))^2+(-12.5)$

$y=a(x+1.5)^2-12.5$ ......(2)

Now, The y-intercept is (0,-8)

Substitute in equation (2),

$-8=a(0+1.5)^2-12.5$

$-8+12.5=a(1.5)^2$

$4.5=2.25a$

$a=\frac{4.5}{2.25}$

$a=2$

Put back in equation (2),

$y=2(x+1.5)^2-12.5$

To find the x-intercept put y=0 in above equation,

$0=2(x+1.5)^2-12.5$

$0=2(x^2+3x+2.25)-12.5$

$0=2x^2+6x+4.5-12.5$

$0=2x^2+6x-8$

$x^2+3x-4=0$

Solve by middle term split,

$x^2+4x-x-4=0$

$x(x+4)-1(x+4)=0$

$(x+4)(x-1)=0$

$x+4=0,x-1=0$

$x=-4,1$

Therefore, The x-intercept are x=-4,1.

Answer 2

The general equation of a parabola is y=ax^2+bx+c  At the y-intercept, x=0 and y= -8:  -8 = a(0)^2 + b(0) + c.  Thus, c = -8.  So, our equation becomes

y = ax^2 + bx - 8.  Next, substitute -1.5 for x and -12.5 for y.  Then,
-12.5 = a(-1.5)^2 + b(-1.5) - 8.  This simplifies to -4.5 = a(2.25) - 1.5b.

Next, take advantage of the info that the vertex is at x= -1.5.
The formula for the vertex is x=-b/(2a).    Letting this formula = -1.5, 

-1.5 = -b/(2a).  We can then solve for b:  1.5 = b/(2a), or 3a = b.

Now go back to the equation we derived previously:  -4.5 = a(2.25) - 1.5b.
Substitute 3a for b:

-4.5 = a(2.25) - 1.5(3a).    Then -4.5 = -2.25a, and a = 4.5/2.25 = 2.

Last, substitute a = 2 into   3a=b to determine the value of b.

b=3(2) = 6.

Therefore, your equation is y=2x^2 + 6x - 8.

Check this result.  Substitute the coordinates of the vertex (-1.5,-12.5) into this equation.  Is the equation still true?  If so, your equation correctly represents this parabola.