Question

consider the function $f(x) = -2x^2 + 12x - 10$.
the vertex of $f(x)$ is $(a, b)$, where $a = \square$ and $b = \square$.
the discriminant of $f(x)$ is $\square$.
note: your answers should be integers.

Explanation:

Step1: Find the x-coordinate of the vertex (A)

For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 12 \). So, \( A = -\frac{12}{2\times(-2)} = -\frac{12}{-4} = 3 \).

Step2: Find the y-coordinate of the vertex (B)

Substitute \( x = 3 \) into the function \( f(x) = -2x^2 + 12x - 10 \). So, \( f(3) = -2\times(3)^2 + 12\times3 - 10 = -2\times9 + 36 - 10 = -18 + 36 - 10 = 8 \). Thus, \( B = 8 \).

Step3: Find the discriminant

The discriminant of a quadratic \( ax^2 + bx + c \) is \( D = b^2 - 4ac \). Here, \( a = -2 \), \( b = 12 \), \( c = -10 \). So, \( D = (12)^2 - 4\times(-2)\times(-10) = 144 - 80 = 64 \).

Answer:

For the vertex: \( A = 3 \), \( B = 8 \)
For the discriminant: \( 64 \)