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Part 1
Complete the square of the given quadratic expression.​ Then, graph the function using the technique of shifting.
f left parenthesis x right parenthesis equals 3 x squared minus 42 x plus 149


To solve this problem, we will break it into two parts: completing the square and graphing the function using shifting.

### Part 1: Completing the Square

We are given the quadratic function:

\[
f(x) = 3x^2 - 42x + 149
\]

#### Step 1: Factor out the coefficient of \(x^2\)
First, factor out the coefficient of \(x^2\) from the first two terms:

\[
f(x) = 3(x^2 - 14x) + 149
\]

#### Step 2: Complete the square
To complete the square, take half of the coefficient of \(x\) in the expression inside the parentheses, square it, and add it inside the parentheses. The coefficient of \(x\) is \(-14\), so:

\[
\frac{-14}{2} = -7
\]
\[
(-7)^2 = 49
\]

Add and subtract \(49\) inside the parentheses:

\[
f(x) = 3(x^2 - 14x + 49 - 49) + 149
\]

This simplifies to:

\[
f(x) = 3((x - 7)^2 - 49) + 149
\]

#### Step 3: Simplify the expression
Now, distribute the \(3\) and combine like terms:

\[
f(x) = 3(x - 7)^2 - 3 \times 49 + 149
\]
\[
f(x) = 3(x - 7)^2 - 147 + 149
\]
\[
f(x) = 3(x - 7)^2 + 2
\]

So, the quadratic expression in completed square form is:

\[
f(x) = 3(x - 7)^2 + 2
\]

### Part 2: Graphing using shifting

The function \(f(x) = 3(x - 7)^2 + 2\) is a transformed version of the basic quadratic function \(y = x^2\), with the following transformations:
- **Horizontal shift**: The term \((x - 7)\) means the graph is shifted **7 units to the right**.
- **Vertical shift**: The constant \(+2\) means the graph is shifted **2 units upwards**.
- **Vertical stretch**: The coefficient \(3\) causes a **vertical stretch** by a factor of 3, making the parabola narrower.

### Graphing details:
1. The vertex of the parabola is at \((7, 2)\).
2. The parabola opens upwards since the coefficient of \((x - 7)^2\) is positive.
3. The axis of symmetry is the vertical line \(x = 7\).
4. The graph is a narrower parabola due to the vertical stretch factor of 3.

Would you like a graphical visualization or further details on this method?

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Here are 5 related questions to deepen your understanding:
1. How does changing the coefficient of \(x^2\) affect the shape of a parabola?
2. What are the steps to find the vertex of a quadratic function in general form?
3. How does completing the square help in solving quadratic equations?
4. Can you explain the relationship between the vertex form and standard form of a quadratic equation?
5. How do different transformations (shifts, stretches, compressions) affect the graph of a quadratic function?

**Tip**: Completing the square is not only useful for graphing, but also for solving quadratic equations and integrating certain functions in calculus.

Complete the square of the given quadratic expression. Then, graph the function using the technique of shifting. f(x) = 3x^2 - 42x + 149

Master the process of completing the square for the quadratic expression f(x) = 3x^2 - 42x + 149. Learn how to graph the function using shifting techniques, covering key concepts such as quadratic transformations, horizontal and vertical shifts, and completing the square. Suitable for students in Grades 9-11.

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