Question

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

Short answer

Complete the square to get \( f(x) = (x + 1)^2 - 1 \) with vertex at \( (-1, -1) \). Graph opens upwards.

Step-by-step solution

- Identify constants

Given the quadratic function \( f(x) = x^2 + 2x \), identify the coefficients. Here, \( a = 1 \), \( b = 2 \).

- Find the term to complete the square

To complete the square, take the coefficient of \( x \) (which is 2), divide by 2, and square it: \( \left( \frac{2}{2} \right)^2 = 1 \).

- Form the perfect square trinomial

Add and subtract the square term inside the function: \( f(x) = x^2 + 2x + 1 - 1 \). This can be written as \( f(x) = (x + 1)^2 - 1 \).

- Rewrite the function

Rewrite the function in its vertex form: \( f(x) = (x + 1)^2 - 1 \). From this, the vertex is found to be \( (-1, -1) \).

- Graph the function

To graph \( f(x) \), plot the vertex \( (-1, -1) \) and note that the parabola opens upwards (since the leading coefficient is positive). Plot additional points if needed and draw the parabola.

Key concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that makes them easier to graph and understand. It involves creating a perfect square trinomial from the quadratic expression. To complete the square for the function \( f(x) = x^2 + 2x \), follow these steps:

- Identify the coefficient of \( x \) (which is 2 in this case).
- Divide it by 2 and square the result: \( \left( \frac{2}{2} \right)^2 = 1 \).
- Add and subtract this square term from the function to form a perfect square: \( f(x) = x^2 + 2x + 1 - 1 \).

This creates a perfect square trinomial plus a constant term, which can then be factored and rewritten in a simpler form.

Vertex Form

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \). This form provides an easy way to identify the vertex of the parabola, which is the point \((h, k)\).

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

- The expression inside the parenthesis, \( (x + 1) \), indicates a horizontal shift. Here, it's \( x + 1 = 0 \), so the vertex is shifted to the left by 1 unit (meaning \( h = -1 \)).
- The constant term \( -1 \) indicates a vertical shift down by 1 unit (so \( k = -1 \)).

Thus, the vertex of the parabola is at \( (-1, -1) \). The function in vertex form makes it straightforward to determine these shifts and plot the vertex.

Graphing Parabolas

Graphing a parabola involves plotting points and understanding the shape and direction of the curve. Starting with the vertex helps set a solid foundation for the graph.

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

- The vertex \( (-1, -1) \) is the minimum point of the parabola, as it opens upwards (since the leading coefficient is positive).
- To graph the parabola, plot the vertex first.
- Additional points can be found by selecting values for \( x \) and substituting them into the function to find corresponding \( y \)-values.

For example:

• If \( x = 0 \), then \( f(0) = (0 + 1)^2 - 1 = 1 - 1 = 0 \). So, the point \( (0, 0) \) is on the graph.
• If \( x = -2 \), then \( f(-2) = (-2 + 1)^2 - 1 = (-1)^2 - 1 = 0 \). So, the point \( (-2, 0) \) is on the graph.

Connect these points to form the parabola. Remember, the shape should be symmetrical about the vertex.