Questions: f(x)=2 x^2+12 x-17 f(x)=-2 x^2-12 x-19

Solution Steps

The given functions are:

• \( f(x) = 2x^2 + 12x - 17 \)
• \( f(x) = -2x^2 - 12x - 19 \)

For a quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \).

• \( a = 2 \)
• \( b = 12 \)
• \( x = -\frac{12}{2 \cdot 2} = -\frac{12}{4} = -3 \)

Substitute \( x = -3 \) back into the function to find the y-coordinate: \[ f(-3) = 2(-3)^2 + 12(-3) - 17 = 2(9) - 36 - 17 = 18 - 36 - 17 = -35 \]

So, the vertex is \( (-3, -35) \).

• \( a = -2 \)
• \( b = -12 \)
• \( x = -\frac{-12}{2 \cdot -2} = -\frac{12}{-4} = 3 \)

Substitute \( x = 3 \) back into the function to find the y-coordinate: \[ f(3) = -2(3)^2 - 12(3) - 19 = -2(9) - 36 - 19 = -18 - 36 - 19 = -73 \]

So, the vertex is \( (3, -73) \).

The direction of a parabola is determined by the sign of the coefficient \( a \) in the quadratic function \( f(x) = ax^2 + bx + c \).

• \( a = 2 \) (positive), so the parabola opens upwards.

• \( a = -2 \) (negative), so the parabola opens downwards.

Final Answer