The given function is:

\[
f(x) = x^2 - 8x + 16
\]

### Step 1: Simplifying the Function
This quadratic expression can be simplified further by recognizing it as a perfect square:

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

This is because:

\[
(x - 4)^2 = x^2 - 8x + 16
\]

### Step 2: Analyzing the Function
- **Vertex form:** The function is now in vertex form, \( f(x) = (x - 4)^2 \), where the vertex of the parabola is at \( (4, 0) \).
- **Axis of symmetry:** The line of symmetry is \( x = 4 \).
- **Minimum value:** Since the square of a number is always non-negative, the minimum value of \( f(x) \) occurs at \( x = 4 \), where \( f(x) = 0 \).

### Step 3: Graphing Insights
- The parabola opens upwards (since the coefficient of \( x^2 \) is positive).
- The graph touches the x-axis at \( x = 4 \), indicating that the only root of the equation \( f(x) = 0 \) is \( x = 4 \).

Would you like more details on graphing this function or any further analysis?

---

Here are 5 related questions:
1. How do you find the vertex of a quadratic function in standard form?
2. What is the importance of recognizing a perfect square trinomial?
3. How does the graph of \( f(x) = x^2 \) compare to \( f(x) = (x-4)^2 \)?
4. Can you factor non-perfect square quadratic functions similarly?
5. How do transformations affect the graph of a quadratic function?

**Tip:** The vertex form of a quadratic equation helps you easily identify the vertex and the direction in which the parabola opens.

Explore how to solve and analyze the quadratic function f(x) = x² - 8x + 16. This problem involves simplifying the expression to its vertex form, identifying key properties of the parabola, and understanding the vertex and symmetry. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation