For the quadratic function f (x) = 4x2 − 4x + 2, answer the following questions.

Sure! Let’s start analyzing the quadratic function \( f(x) = 4x^2 - 4x + 2 \). Here’s how we can approach answering various questions related to this function:

### 1. **Determine the vertex of the quadratic function.**

The vertex form of a quadratic function is given by:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex.

We can find the vertex by using the formula for the x-coordinate of the vertex:
\[
x = \frac{-b}{2a}
\]
where \( a = 4 \) and \( b = -4 \) from the given function.

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

Now, substitute \( x = \frac{1}{2} \) back into the function to find the y-coordinate:
\[
f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^2 - 4\left(\frac{1}{2}\right) + 2
\]
\[
f\left(\frac{1}{2}\right) = 4\left(\frac{1}{4}\right) - 2 + 2 = 1
\]

Thus, the vertex is \(\left( \frac{1}{2}, 1 \right)\).

### 2. **Find the axis of symmetry.**

The axis of symmetry is the vertical line that passes through the vertex. Since the x-coordinate of the vertex is \( \frac{1}{2} \), the axis of symmetry is:
\[
x = \frac{1}{2}
\]

### 3. **Determine whether the function has a minimum or maximum value.**

Since the coefficient of \(x^2\) (i.e., \( a = 4 \)) is positive, the parabola opens upwards. This means the function has a **minimum value** at the vertex.

### 4. **Find the minimum or maximum value.**

The minimum value of the function is the y-coordinate of the vertex, which we calculated as \(1\). Therefore, the minimum value of the function is:
\[
f(x)_{\text{min}} = 1
\]

### 5. **Find the y-intercept.**

To find the y-intercept, set \(x = 0\) and solve for \(f(0)\):
\[
f(0) = 4(0)^2 - 4(0) + 2 = 2
\]
Thus, the y-intercept is \( (0, 2) \).

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Let me know if you'd like any more details or clarifications! Here are five related questions:

1. How do we convert the quadratic function into vertex form?
2. What are the x-intercepts of the function \( f(x) = 4x^2 - 4x + 2 \)?
3. Can you graph the function based on the vertex, axis of symmetry, and intercepts?
4. How does the discriminant of a quadratic equation relate to the nature of its roots?
5. How do changes to the coefficient \(a\) in \(ax^2 + bx + c\) affect the shape of the parabola?

**Tip:** The vertex form of a quadratic function can be useful for quickly identifying the vertex and graphing the function.

For the quadratic function f(x) = 4x^2 − 4x + 2, answer the following questions.

Explore the detailed analysis of the quadratic function f(x) = 4x^2 - 4x + 2, including finding the vertex, axis of symmetry, and minimum value. Suitable for students in Grades 9-10.

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