To analyze the quadratic function $f(x) = -2x^2 + 8x - 12$, let's go through each part of the problem.

Part 1: Finding the Vertex

The vertex form of a quadratic function $f(x) = ax^2 + bx + c$ has its vertex at the point: $\left( x, y \right) = \left( \frac{-b}{2a}, f\left( \frac{-b}{2a} \right) \right)$

For $f(x) = -2x^2 + 8x - 12$, we have:

• $a = -2$
• $b = 8$

1. Calculate $x$-coordinate of the vertex: $x = \frac{-b}{2a} = \frac{-8}{2 \cdot -2} = \frac{-8}{-4} = 2$

2. Substitute $x = 2$ into $f(x)$ to find the $y$-coordinate of the vertex: $f(2) = -2(2)^2 + 8(2) - 12 = -2 \cdot 4 + 16 - 12 = -8 + 16 - 12 = -4$

Thus, the vertex is at: $\text{Vertex} = (2, -4)$

Part 2: Graphing the Function

The graph of this quadratic function will be a downward-opening parabola, as the leading coefficient $a = -2$ is negative.

Part 3: Axis of Symmetry

The axis of symmetry for a quadratic function is the vertical line passing through the vertex. So the axis of symmetry here is: $x = 2$

Part 4: Domain

For any quadratic function, the domain is all real numbers because $x$ can take any value. Thus: $\text{Domain} = (-\infty, \infty)$

Part 5: Range

Since the parabola opens downward, the function has a maximum value at the vertex. The maximum $y$-value is the $y$-coordinate of the vertex, which is $-4$. Thus: $\text{Range} = (-\infty, -4]$

Summary of Answers

• (a) Vertex: $(2, -4)$
• (b) Axis: $x = 2$
• (c) Domain: $(-\infty, \infty)$
• (d) Range: $(-\infty, -4]$

Would you like a detailed graph of this function?

Relative Questions

1. How do you determine the direction a parabola opens based on the coefficient $a$?
2. What is the significance of the vertex in a quadratic function?
3. How does the value of $b$ affect the location of the axis of symmetry?
4. How would the range change if the parabola opened upwards?
5. Can you have a different domain for a quadratic function under any circumstances?

Tip

Remember, for any quadratic function, the axis of symmetry will always be $x = \frac{-b}{2a}$, dividing the parabola into two symmetrical halves.