A quadratic equation has a vertex at (−3/2,7/2) and a y-intercept of 8.
Which equations could describe this quadratic equation?
There is more than one correct answer. Select all that apply.
a) y=(x+32)2+74
b) y=x2+3x+4
c) y=2(x+32)2+74
d) y=x2−3x+7
e) y=2((x+32)2+74)
f) y=2x2+6x+8
given: vertex is (−3/2,7/2) and a y-intercept of 8
so the equation is
y = a(x+3/2)^2 + 7/2
but (0,8) lies on it
8 = a(9/4) + 7/2
times 4
32 = 9a + 14
9a = 18
a = 2
so we have y = 2(x + 3/2)^2 + 7/2
expand and simplify, then see if you find the match
(I found it)
To find the equation of a quadratic function with vertex form (h, k) and y-intercept (0, b), we can use the standard form of a quadratic equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex and a is a constant determining the shape of the parabola.
In this case, the vertex is (-3/2, 7/2) and the y-intercept is (0, 8).
Plugging these values into the equation, we get:
y = a(x - (-3/2))^2 + 7/2
= a(x + 3/2)^2 + 7/2
Since the y-intercept is (0, 8), we can substitute x=0 and y=8 into the equation to find the value of "a":
8 = a(0 + 3/2)^2 + 7/2
= a(3/2)^2 + 7/2
= a(9/4) + 7/2
= (9a + 14)/4
Simplifying the equation, we get:
9a + 14 = 32
9a = 18
a = 2
Now we have the value of "a" which is 2.
Plugging this value back into the equation, we can write the equation of the quadratic function as:
y = 2(x + 3/2)^2 + 7/2
Expanding and simplifying the equation further, we get:
y = 2(x^2 + 3x + 9/4) + 7/2
= 2x^2 + 6x + 18/4 + 7/2
= 2x^2 + 6x + 9/2 + 14/2
= 2x^2 + 6x + 23/2
Therefore, the equation that could describe this quadratic equation is:
f) y = 2x^2 + 6x + 23/2
None of the other given equations match the given information.
To find the equation of a quadratic function given its vertex and y-intercept, we can use the vertex form of a quadratic function, which is y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
In this case, the vertex is (-3/2, 7/2), which means h = -3/2 and k = 7/2.
Now we have the equation y = a(x-(-3/2))^2 + 7/2, which simplifies to y = a(x+3/2)^2 + 7/2.
To find the value of a, we need to use the y-intercept, which is 8. Substituting x = 0 and y = 8 into the equation, we get:
8 = a(0+3/2)^2 + 7/2
8 = a(3/2)^2 + 7/2
8 = 9a/4 + 7/2
8 - 7/2 = 9a/4
16/2 - 7/2 = 9a/4
9/2 = 9a/4
(9/2) * (4/9) = a
2 = a
Now we know that a = 2, so the equation becomes y = 2(x+3/2)^2 + 7/2.
Let's test each option to see which ones satisfy the given conditions.
a) y = (x+32)^2 + 74
This equation is not in vertex form. It is unlikely to be the correct answer.
b) y = x^2 + 3x + 4
This equation is not in vertex form either. It is unlikely to be the correct answer.
c) y = 2(x+32)^2 + 74
This equation is in vertex form. Let's check if it satisfies the given conditions.
Vertex: (-3/2, 7/2) => (h, k) = (-3/2, 7/2)
y-intercept: 8
Substitute x = -3/2 into the equation:
y = 2(-3/2+3/2)^2 + 74
y = 2(0)^2 + 74
y = 74
The vertex is correct, but the y-intercept is not 8. Therefore, this equation does not satisfy the conditions.
d) y = x^2 - 3x + 7
This equation is not in vertex form. It is unlikely to be the correct answer.
e) y = 2((x+32)^2 + 74)
This equation is not in vertex form either. It is unlikely to be the correct answer.
f) y = 2x^2 + 6x + 8
This equation is not in vertex form. It is unlikely to be the correct answer.
After checking all the options, we find that none of the given equations satisfy the conditions of having a vertex at (-3/2, 7/2) and a y-intercept of 8. Therefore, there are no correct answers among the given options.