Questions: For the quadratic function f(x)=x^2+2x, answer parts (a) through (f). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.

For the quadratic function f(x)=x^2+2x, answer parts (a) through (f).
(a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.

Transcript text: For the quadratic function $f(x)=x^{2}+2 x$, answer parts (a) through (f). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.

Solution

Solution Steps

Solution Approach

To find the vertex of the quadratic function $ f(x) = x^2 + 2x $, we can use the vertex formula for a quadratic function in the form $ ax^2 + bx + c $, which is given by $ x = -\frac{b}{2a} $. Once we find the x-coordinate of the vertex, we can substitute it back into the function to find the y-coordinate. The axis of symmetry is the vertical line that passes through the vertex, given by $ x = -\frac{b}{2a} $. Since the coefficient of $ x^2 $ (which is 1) is positive, the graph is concave up.

Step 1: Find the Vertex

To find the vertex of the quadratic function $ f(x) = x^2 + 2x $, we use the vertex formula $ x = -\frac{b}{2a} $. Here, $ a = 1 $ and $ b = 2 $. Thus, we calculate:

\[ x = -\frac{2}{2 \cdot 1} = -1 \]

Next, we substitute $ x = -1 $ back into the function to find the y-coordinate of the vertex:

\[ y = f(-1) = (-1)^2 + 2(-1) = 1 - 2 = -1 \]

Therefore, the vertex is $ (-1, -1) $.

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function is given by the line $ x = -\frac{b}{2a} $. From our previous calculation, we have:

\[ \text{Axis of symmetry: } x = -1 \]

Step 3: Determine Concavity

The concavity of the graph is determined by the coefficient $ a $ in the quadratic function. Since $ a = 1 $ (which is positive), the graph is concave up.