Questions: What is the vertex for the parabola defined by (y=frac12(x-4)(x+2)) ? (0,-4) (-1,-2.5) (1,-4.5) (4,0)

Solution Steps

To find the vertex of the parabola defined by the equation \( y = \frac{1}{2}(x-4)(x+2) \), we need to convert the equation to its vertex form or use the properties of parabolas. The vertex form of a parabola is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. Alternatively, we can find the x-coordinate of the vertex using the formula \( x = \frac{-b}{2a} \) for the standard form \( y = ax^2 + bx + c \).

1. Expand the given equation to standard form \( y = ax^2 + bx + c \).
2. Use the formula \( x = \frac{-b}{2a} \) to find the x-coordinate of the vertex.
3. Substitute the x-coordinate back into the equation to find the y-coordinate of the vertex.

The given equation of the parabola is

\[ y = \frac{1}{2}(x - 4)(x + 2) \]

Expanding this, we get

\[ y = 0.5x^2 - 1.0x - 4.0 \]

From the expanded equation \(y = 0.5x^2 - 1.0x - 4.0\), we identify the coefficients:

• \(a = 0.5\)
• \(b = -1.0\)
• \(c = -4.0\)

To find the x-coordinate of the vertex, we use the formula

\[ x = \frac{-b}{2a} \]

Substituting the values of \(b\) and \(a\):

\[ x = \frac{-(-1.0)}{2 \cdot 0.5} = \frac{1.0}{1.0} = 1.0 \]

Next, we substitute \(x = 1.0\) back into the equation to find the y-coordinate:

\[ y = 0.5(1.0)^2 - 1.0(1.0) - 4.0 = 0.5 - 1.0 - 4.0 = -4.5 \]

Thus, the vertex of the parabola is

\[ (1.0, -4.5) \]

Final Answer