(1-7) write each function in vertex form (8-10) write each funtion in standard form?
1) Y=x^2+3x-10
2) Y=x^2-9x
3) Y=X^2+x
4) Y=x^2+5x+4
5) Y=4x^2+8x-3
6) Y=(3/4)x^2+9x
7) Y=-2x^2+2x+1
8) Y=(x-3)^2+1
9) Y=2(x-1)^2-3
10)Y=-3(x+4)^2+1
1. Y = x^2+3x-10.
Y = a(x-h)^2 + k.
h = Xv = -b/2a = -3/2.
k = (-3/2)^2 + 3(-3/2) - 10 = -49/4.
V(h,k).
V(-3/2,-49/4).
Y = 1(X+3/2)^2 - 49/4.
2. Y = x^2 - 9x.
Y = a(X-h)^2 + k.
h = Xv = -b/2a = 9/2.
k = (9/2)^2- 9(9/2) = -81/4.
V(h,k) = (9/2,-81/4).
Y = 1(X-9/2)^2 - 81/4.
8. Y = (x-3)^2 + 1.
Y = x^2 _ 6x + 9 + 1.
Y = X^2 - 6x + 10.
The remaining problems are similar to
those I've solved.
five less than a number is at least-28
To write each function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Let's go step by step for each of the functions:
1) Y = x^2 + 3x - 10
To complete the square, we add and subtract the square of half the coefficient of x (3/2)^2 = 9/4:
Y = (x^2 + 3x + 9/4) - 9/4 - 10
= (x^2 + 3x + 9/4) - 49/4
Now we can write it in vertex form:
Y = (x + 3/2)^2 - 49/4
Vertex Form: Y = (x + 3/2)^2 - 49/4
2) Y = x^2 - 9x
To complete the square, we add and subtract the square of half the coefficient of x (-9/2)^2 = 81/4:
Y = (x^2 - 9x + 81/4) - 81/4
Now we can write it in vertex form:
Y = (x - 9/2)^2 - 81/4
Vertex Form: Y = (x - 9/2)^2 - 81/4
3) Y = x^2 + x
To complete the square, we add and subtract the square of half the coefficient of x (1/2)^2 = 1/4:
Y = (x^2 + x + 1/4) - 1/4
Now we can write it in vertex form:
Y = (x + 1/2)^2 - 1/4
Vertex Form: Y = (x + 1/2)^2 - 1/4
4) Y = x^2 + 5x + 4
To complete the square, we add and subtract the square of half the coefficient of x (5/2)^2 = 25/4:
Y = (x^2 + 5x + 25/4) - 25/4 + 4
= (x^2 + 5x + 25/4) - 9/4
Now we can write it in vertex form:
Y = (x + 5/2)^2 - 9/4
Vertex Form: Y = (x + 5/2)^2 - 9/4
5) Y = 4x^2 + 8x - 3
First, let's factor out the common factor of 4:
Y = 4(x^2 + 2x) - 3
To complete the square, we add and subtract the square of half the coefficient of x (2)^2 = 4:
Y = 4(x^2 + 2x + 4) - 3 - 16
= 4(x^2 + 2x + 4) - 19
Now we can write it in vertex form:
Y = 4(x + 1)^2 - 19
Vertex Form: Y = 4(x + 1)^2 - 19
6) Y = (3/4)x^2 + 9x
First, let's factor out the common factor of (3/4):
Y = (3/4)(x^2 + 12x)
To complete the square, we add and subtract the square of half the coefficient of x (12/2)^2 = 36:
Y = (3/4)(x^2 + 12x + 36) - (3/4) * 36
Now we can write it in vertex form:
Y = (3/4)(x + 6)^2 - 27
Vertex Form: Y = (3/4)(x + 6)^2 - 27
7) Y = -2x^2 + 2x + 1
We can factor out a common factor of -1:
Y = -(2x^2 - 2x - 1)
To complete the square, we add and subtract the square of half the coefficient of x (-2/2)^2 = 1:
Y = -(2(x^2 - x + 1/4) - 1/2) - 1
Now we can write it in vertex form:
Y = -2(x - 1/2)^2 + 1/2
Vertex Form: Y = -2(x - 1/2)^2 + 1/2
Now let's write each function in standard form:
8) Y = (x - 3)^2 + 1
Standard Form: Y = x^2 - 6x + 10
9) Y = 2(x - 1)^2 - 3
Standard Form: Y = 2x^2 - 4x + 1
10) Y = -3(x + 4)^2 + 1
Standard Form: Y = -3x^2 - 24x - 47