Question

Graph the quadratic y=x^2+2x+4
. What is the axis of symmetry?

Answer 1

To graph the quadratic function \( y = x^2 + 2x + 4 \), we can first identify its vertex and axis of symmetry.

### Step 1: Identify the Axis of Symmetry

The formula for the axis of symmetry for a quadratic equation of the form \( y = ax^2 + bx + c \) is given by:

\[
x = -\frac{b}{2a}
\]

In our equation, \( a = 1 \) and \( b = 2 \).

Plugging in the values:

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

So, the axis of symmetry is \( x = -1 \).

### Step 2: Find the Vertex

To find the vertex, we substitute \( x = -1 \) back into the quadratic equation:

\[
y = (-1)^2 + 2(-1) + 4
\]
\[
y = 1 - 2 + 4 = 3
\]

Thus, the vertex of the parabola is at the point \( (-1, 3) \).

### Step 3: Determine Additional Points for Graphing

To graph the quadratic, we can find some additional points:

1. For \( x = -2 \):
\[
y = (-2)^2 + 2(-2) + 4 = 4 - 4 + 4 = 4 \quad \text{(point: } (-2, 4))
\]

2. For \( x = 0 \):
\[
y = (0)^2 + 2(0) + 4 = 4 \quad \text{(point: } (0, 4))
\]

3. For \( x = 1 \):
\[
y = (1)^2 + 2(1) + 4 = 1 + 2 + 4 = 7 \quad \text{(point: } (1, 7))
\]

4. For \( x = -3 \):
\[
y = (-3)^2 + 2(-3) + 4 = 9 - 6 + 4 = 7 \quad \text{(point: } (-3, 7))
\]

### Step 4: Sketch the Graph

Now, we can sketch the graph using the points we found:

- Vertex: \( (-1, 3) \)
- Points: \( (-2, 4), (0, 4), (1, 7), (-3, 7) \)

Given these points, we can see the general shape of the parabola, which opens upwards since \( a > 0 \).

### Summary

- **Axis of symmetry:** \( x = -1 \)
- **Vertex:** \( (-1, 3) \)

You can use this information to draw the graph of the function on a coordinate plane. The parabola will be symmetric about the line \( x = -1 \) and will have its vertex point at \( (-1, 3) \).