Question

Complete the square of each quadratic expression. Then graph each function using graphing techniques. $$ f(x)=3 x^{2}+6 x+1 $$

Short answer

Completed square form is f(x) = 3(x + 1)^2 - 2. The vertex is (-1, -2) and the parabola opens upwards.

Step-by-step solution

- Factor out the coefficient of the quadratic term

Take the coefficient of the quadratic term outside of the parentheses: f(x) = 3(x^2 + 2x) + 1

- Complete the square inside the parentheses

To complete the square: 1. Take half of the coefficient of the x term (which is 2), giving you 1. 2. Square it to get 1. 3. Add and subtract this square inside the parentheses: f(x) = 3(x^2 + 2x + 1 - 1) + 1

- Simplify the expression

Simplify the expression by combining like terms inside the parentheses: f(x) = 3((x + 1)^2 - 1) + 1

- Distribute the 3 and simplify further

Distribute the 3 and simplify: f(x) = 3(x + 1)^2 - 3 + 1 f(x) = 3(x + 1)^2 - 2

- Identify the vertex form of the quadratic

The simplified form now looks like the standard vertex form: f(x) = 3(x + 1)^2 - 2 Here, the vertex is at (-1, -2)

- Graph the function

To graph the function: 1. Plot the vertex at (-1, -2). 2. Use the coefficient 3 to determine the direction and width of the parabola. Since 3 is positive, the parabola opens upwards and is narrower than the standard parabola y = x^2. 3. Sketch the parabola using these characteristics.

Key concepts

quadratic functions

Quadratic functions are a type of polynomial function where the highest-degree term is squared. The general form is given by:
\[\begin{equation} f(x) = ax^2 + bx + c \end{equation}\]
Here, 'a', 'b', and 'c' are constants, with 'a' not equal to zero. These functions graph as parabolas, which can either open upwards if 'a' is positive or downwards if 'a' is negative.
Understanding the structure of quadratic functions is crucial as it helps in identifying key elements such as the vertex (the highest or lowest point on the parabola), the axis of symmetry, and the y-intercept.

Different forms of quadratic equations, like the standard form and the vertex form, serve various purposes in graphing and solving quadratic equations.

vertex form

The vertex form of a quadratic function provides a straightforward way to identify the vertex of the parabola. The vertex form is written as:
\[\begin{equation} f(x) = a(x - h)^2 + k \end{equation}\]
Here, \( h, k \) indicates the vertex of the parabola. Finding the vertex form often involves a process called 'completing the square'.
To convert a quadratic function from its general form, \( f(x) = ax^2 + bx + c \), to the vertex form:

• Factor out the coefficient 'a' from the x-terms.
• Complete the square within the parentheses by adding and subtracting the same value inside.
• Simplify the expression to isolate the square term.

In our exercise, we transformed \( 3x^2 + 6x + 1 \) into \( 3(x + 1)^2 - 2 \) by completing the square. This helps us easily identify that the vertex is at \(-1, -2\).

graphing techniques

Graphing a quadratic function becomes easier with the knowledge of its vertex form. Follow these simple steps to sketch the graph:

• Identify and plot the vertex on the coordinate plane.
• Determine the direction of the parabola using the leading coefficient 'a'. If 'a' is positive, the parabola opens upwards; if negative, it opens downwards.
• Check the width of the parabola: A larger absolute value of 'a' makes the parabola narrower, and a smaller absolute value of 'a' makes it wider compared to the basic parabola \(y = x^2\).

For our function \( f(x) = 3(x + 1)^2 - 2 \):

• The vertex is at (-1, -2).
• The coefficient 3 suggests that the parabola opens upwards and is narrower than \( y = x^2 \).

By following these steps and understanding these concepts, you can graph any quadratic function with confidence!