Integrated math 2

Question

Find the vertex, equation, use the point:(4,4), solve for a:, let b=1, so the equation is: answer:

Answer 1

There are infinitely many parabolas passing through (4,4) such that b=1.

y = ax² + x + c

4 = 16a +4 +c

a = -c/16

As long as the value of a is -1/16 or the value of c the conditions are satisfied.

.Check is out here, using the slider on c to see some of the infinitely many parabolas passing through (4,4) such that the value of b is 1.

desmos.com/calculator/uiazpypo21

Note that if c = 0, there is a line passing through (4,4) instead of a parabola.

Answer 2

y= ax^2 +x + c = 0 in standard form

4= 16a +4 + c = 0

16a = -c

a=-c/16

y = a(x+h)^2+ k in vertex form

4 = a(16+2(4h)+ h^2) + k, b=2h. h=1/2

4= 16a + 8ah + ah^2 + k

16a + 8ah+ah^2 +k -4 = 0 c= k-4 +ah^2 = k-4- ch^2/16,

c+ch^2/16 = k-4

c = (k-4)/(1+h^2/16)= (k-4)(65)/64= 65k/64 - 65/16

a=65/256 - 65k/1074

y= (-c/16)x^2 +x + (k+4)/(1+h^2/16)

4= - c+ 4 +(k+4)/(1+h^2/16)

y = (65/256 - 65k/1074)(x- 1/2)^2 - k

4= (65/256- 65k/1074)(4-1/2)^2 - k one equation one unknown, solve for k

then use the k value to solve for c then b or just complete the vertex

(h,k) is the vertex = (1/2, k)

axis of symmetry is x= 1/2

if (4,4) is a point on the graph so is (-3,4)

it's a vertical parabola, downward opening if a is negative, upward opening if a>0

4= (65/256 -65k/1074)(49/4) - k

k+4 = 65/1074 - 65k/4296

k(1+65/4296 = 65/1074 -4, k<0, so parabola opens downward and a is negative

k = (-4(1074)+65)/1074 over 2361/4296

= -4231/1074)/(2361/4296) = -(4231)(4296)/(1074)(2361)

vertex = (1/2, -4231(4296)/1074(2361)) = (1/2, -4231(2148)/537(2361)

no guarantee this is error free. any calculations this tedious just might have a few errrors

may help to sketch the parabola points, axis of symmetry, c=y intercept

let y=0 then the x values are the x intercepts