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