https://file.aiquickdraw.com/mathbot/file/1728333363047jx5zmpem.jpg

Let's break down the given quadratic function and answer the required parts step by step.

The function given is:

\[
f(x) = -2x^2 + 2x - 1
\]

This is a quadratic function with a downward-opening parabola (because the coefficient of \(x^2\) is negative).

### Finding the Vertex
The vertex form of a quadratic function is given by:

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

For the given function \(f(x) = -2x^2 + 2x - 1\):

- \(a = -2\)
- \(b = 2\)

Substitute the values into the vertex formula:

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

So, the x-coordinate of the vertex is \(x = \frac{1}{2}\).

### (a) Increasing Interval:
Since the parabola opens downward, the function is increasing to the left of the vertex and decreasing to the right. Therefore, the function is increasing on the interval:

\[
\left( -\infty, \frac{1}{2} \right]
\]

### (b) Decreasing Interval:
The function decreases after the vertex. Therefore, the function is decreasing on the interval:

\[
\left[ \frac{1}{2}, \infty \right)
\]

### (f) Determining where \(f(x) > 0\) and \(f(x) < 0\):
To find where the function is positive or negative, we need to solve for the roots of the quadratic equation by using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting \(a = -2\), \(b = 2\), and \(c = -1\) into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{2^2 - 4(-2)(-1)}}{2(-2)} = \frac{-2 \pm \sqrt{4 - 8}}{-4} = \frac{-2 \pm \sqrt{-4}}{-4}
\]

Since the discriminant is negative (\(\sqrt{-4} = 2i\)), this quadratic has no real roots. Therefore, the parabola does not cross the x-axis, and since the parabola opens downward, the function is always negative.

Thus, the correct choice for part (f) is:

**C.** \(f(x) > 0\) on an empty set and \(f(x)\) is never positive.

---

Would you like more details or have any questions on these steps? Here are five related questions to expand your understanding:

1. How can the discriminant help determine the number of real roots for a quadratic equation?
2. How does the leading coefficient affect the direction of a parabola?
3. Can you describe the relationship between the vertex and the intervals of increasing and decreasing behavior for a quadratic function?
4. What are the implications of a quadratic function having no real roots?
5. How do you rewrite a quadratic equation in vertex form?

**Tip:** When solving quadratic equations, always check the discriminant (\(b^2 - 4ac\)) to determine whether the solutions are real or complex.

For the quadratic function f(x) = -2x^2 + 2x - 1, answer parts (a) through (f).

Learn how to determine the increasing and decreasing intervals and roots for the quadratic function f(x) = -2x^2 + 2x - 1. This problem covers key concepts like finding the vertex, using the quadratic formula, and understanding the discriminant. Suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation