y=x^2+bx+7 has a vertex at (-4,-9). What is the value of b?
well, plug in your point!
(-4)^2+b(-4)+7 = -9
16-4b+7 = -9
-4b = -32
b = 8
check:
x^2+8x+7 = (x+4)^2-9
To find the value of b in the given quadratic equation, we can use the fact that the vertex of a quadratic function is of the form (-b/2a, f(-b/2a)).
In this case, we are given that the vertex is (-4, -9), which means that -b/2a = -4 and f(-b/2a) = -9.
The x-coordinate of the vertex, -b/2a, is equal to -4.
Therefore, we have the equation:
-b/2a = -4
Since the coefficient of x^2 is 1, a=1. Plugging in a=1 into the equation, we get:
-b/2(1) = -4
-b/2 = -4
-b = -4(2)
-b = -8
Therefore, the value of b is -8.
To find the value of b, we need to determine the equation of the quadratic function in vertex form. The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex. In this case, the vertex is (-4, -9), so we can replace h with -4 and k with -9:
y = a(x - (-4))^2 + (-9)
y = a(x + 4)^2 - 9
Now we can compare this equation with the given equation y = x^2 + bx + 7 and determine the value of b.
Comparing the two equations, we see that a = 1 (since there is no leading coefficient for x^2 term in the given equation). Thus, we have:
(x + 4)^2 - 9 = x^2 + bx + 7
Expanding the square term, we get:
(x^2 + 8x + 16) - 9 = x^2 + bx + 7
Simplifying further:
x^2 + 8x + 7 = x^2 + bx + 7
Comparing the corresponding coefficients, we can equate the x terms:
8x = bx
Since the x terms are equal, we can conclude that the value of b is 8.
Therefore, the value of b in the equation y = x^2 + bx + 7 is 8.