Let g(x) = -2x2 + bx + c be a quadratic function, defined everywhere, where b and c are constants. If x = 1 marks the location of one of the zeros of this quadratic function, and if the y-intercept of this function is at (0, 5), then use this information to name constant b.
g ( x ) = - 2 * x ^ 2 + b * x + c
For:
x = 0
y= 5
Then:
g( x ) = - 2 * 0 ^ 2 + 0 * x + c = 5
g( x ) = c = 5
c = 5
For:
x= 1
y = 0
g ( x ) = - 2 * 1 ^ 2 + b * 1 + c = 0
g ( x ) = - 2 + b + c = 0
g ( x ) = - 2 + b + 5 = 0
g ( x ) = b + 3 = 0
b = -3
To find the constant b, we need to use the information given.
Let's start with the fact that x = 1 marks the location of one of the zeros of the quadratic function. In other words, when x = 1, g(x) = 0.
Substituting x = 1 into the quadratic function g(x), we have:
g(1) = -2(1)^2 + b(1) + c
Simplifying this equation, we get:
0 = -2 + b + c
Next, we have the y-intercept of the function, which is at the point (0, 5). The y-intercept occurs when x = 0, so we can substitute these values into the quadratic function:
g(0) = -2(0)^2 + b(0) + c
g(0) = 0 + 0 + c
g(0) = c
Therefore, we know that c is equal to 5.
Now let's go back to the equation we obtained earlier:
0 = -2 + b + c
Substituting c = 5, we get:
0 = -2 + b + 5
Simplifying this equation, we have:
0 = b + 3
Rearranging the equation, we find:
b = -3
Hence, the constant b in this quadratic function is -3.