The function given is:

\[ f(x) = 4x^2 - 2x + 12 \]

Let's break down its key features:

### 1. **Type of Function:**
This is a quadratic function, which means its graph will be a parabola. Since the coefficient of \( x^2 \) is positive (4), the parabola opens upwards.

### 2. **Vertex:**
The vertex form of a quadratic function \( ax^2 + bx + c \) can be found using the vertex formula:

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

For \( f(x) = 4x^2 - 2x + 12 \), we have:
- \( a = 4 \)
- \( b = -2 \)

So, the x-coordinate of the vertex is:

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

Now, plug this value back into the function to find the y-coordinate of the vertex:

\[
f\left(\frac{1}{4}\right) = 4\left(\frac{1}{4}\right)^2 - 2\left(\frac{1}{4}\right) + 12
\]
\[
= 4\left(\frac{1}{16}\right) - \frac{1}{2} + 12
\]
\[
= \frac{1}{4} - \frac{1}{2} + 12
\]
\[
= \frac{1 - 2}{4} + 12
\]
\[
= -\frac{1}{4} + 12
\]
\[
= \frac{-1 + 48}{4}
\]
\[
= \frac{47}{4}
\]

So, the vertex is at:

\[
\left( \frac{1}{4}, \frac{47}{4} \right)
\]

### 3. **Axis of Symmetry:**
The axis of symmetry is the vertical line that passes through the vertex, given by:

\[
x = \frac{1}{4}
\]

### 4. **Y-intercept:**
To find the y-intercept, set \( x = 0 \):

\[
f(0) = 4(0)^2 - 2(0) + 12 = 12
\]

So, the y-intercept is at \( (0, 12) \).

### 5. **Range and Domain:**
- **Domain:** The domain of any quadratic function is all real numbers, so \( x \in \mathbb{R} \).
- **Range:** Since the parabola opens upwards and the vertex is the minimum point, the range is \( y \geq \frac{47}{4} \).

### Summary:
- **Vertex:** \( \left( \frac{1}{4}, \frac{47}{4} \right) \)
- **Axis of Symmetry:** \( x = \frac{1}{4} \)
- **Y-intercept:** \( (0, 12) \)
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( \left[ \frac{47}{4}, \infty \right) \)

Would you like more details or have any questions?

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### Related Questions:
1. How do you find the x-intercepts of \( f(x) = 4x^2 - 2x + 12 \)?
2. What is the significance of the discriminant in a quadratic function?
3. How can we convert \( f(x) \) into vertex form?
4. What are the properties of the graph of \( f(x) = 4x^2 - 2x + 12 \)?
5. How does changing the coefficient of \( x^2 \) affect the shape of the parabola?

**Tip:** The discriminant of a quadratic equation, \( b^2 - 4ac \), helps determine the nature of its roots (real or complex).

Explore the detailed analysis of the quadratic function f(x) = 4x² - 2x + 12. Learn how to find its vertex, axis of symmetry, y-intercept, and range. Ideal for students in Grades 9-10 studying algebra and quadratic functions.

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