Use the information provided to write the vertex form equation of each parabola.
a)y + 60x + 294 = -3x^2
b)−(y+1) = (x−4)^2
a. y = -3x^2 -60x - 294.General or STD Form.
h = -b/2a = 60 / -6 = -10.
k = -3(-10)^2 - 60*-10 - 294 = 6.
y = a(x-h)^2 + k,
y = -3(x+10)^2 + 6. Vertex Form.
b. -(y+1) = (x-4)^2,
-y-1 = x^2 - 8x + 16,
-y = x^2 - 8x + 17,
y = -x^2 + 8x - 17,
h = -b/2a = -8 / -2 = 4.
k = -(4)^2 +8*4 - 17 = -1.
y = -(x-4)^2 - 1.
To write the vertex form equation of each parabola, we need to convert the given equations into the form:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
a) For the equation y + 60x + 294 = -3x^2:
Step 1: Move the constant term to the other side of the equation:
y + 60x = -3x^2 - 294
Step 2: Rewrite the equation by completing the square for the x terms:
y + 60x = -3(x^2 + 98x/3)
To complete the square, we take half of the coefficient of x (in this case, 98/3) and square it, which gives us (98/6)^2 = 9604/36.
Step 3: Add and subtract this value inside the parentheses:
y + 60x = -3(x^2 + 98x/3 + 9604/36 - 9604/36)
Simplifying further:
y + 60x = -3(x^2 + 98x/3 + 9604/36) - 294
Step 4: Factor the trinomial inside the parentheses:
y + 60x = -3(x + 98/6)^2 - 294
Step 5: Simplify the equation:
y + 60x = -3(x + 49/3)^2 - 294
Step 6: Rewrite the equation in vertex form:
y = -3(x + 49/3)^2 - 294 - 60x
So, the vertex form equation of parabola (a) is y = -3(x + 49/3)^2 - 60x - 294.
b) For the equation -(y + 1) = (x - 4)^2:
Step 1: Distribute the negative sign to the terms inside the parentheses:
-y - 1 = (x - 4)^2
Step 2: Move the constant term to the other side:
-y = (x - 4)^2 + 1
Step 3: Multiply through by -1 to isolate y:
y = -(x - 4)^2 - 1
So, the vertex form equation of parabola (b) is y = -(x - 4)^2 - 1.