Which could be the graph of `y=x^2 + 3x+k` , where k is an integer ?

Options

A.
B.
C.
D.

Answer

To determine which graph could represent the equation \( y = x^2 + 3x + k \), where \( k \) is an integer, we can follow these steps:

### Step 1: Identify the Type of Graph
The equation \( y = x^2 + 3x + k \) is a quadratic equation in the standard form \( y = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 3 \), and \( c = k \).

### Step 2: Determine the Direction of the Parabola
Since \( a = 1 \) (which is greater than 0), the parabola opens upwards. This means that any graph option that shows a downward-opening parabola can be eliminated.

**Hint:** Remember that the sign of \( a \) determines the direction of the parabola. If \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.

### Step 3: Find the Vertex of the Parabola
The x-coordinate of the vertex of a parabola given by \( y = ax^2 + bx + c \) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
Substituting \( a = 1 \) and \( b = 3 \):
\[ x = -\frac{3}{2 \cdot 1} = -\frac{3}{2} \]

### Step 4: Analyze the Vertex Position
The x-coordinate of the vertex is \( -\frac{3}{2} \), which is negative. This means that the vertex lies in the second quadrant (where x is negative).

**Hint:** The vertex's x-coordinate gives us information about the graph's position. Look for options where the vertex is located in the second quadrant.

### Step 5: Evaluate the Graph Options
Now, we can evaluate the given graph options:
- **Option A:** If the vertex is in the first quadrant (x positive), eliminate this option.
- **Option B:** If the vertex is also in the first quadrant, eliminate this option.
- **Option C:** If the vertex is in the second quadrant (x negative), this option remains.
- **Option D:** If the vertex is in the third quadrant (x negative but opens downwards), eliminate this option.

### Conclusion
After evaluating the options, we find that **Option C** is the only graph that matches the criteria of having an upward-opening parabola with a vertex at \( x = -\frac{3}{2} \).

**Final Answer:** The graph of \( y = x^2 + 3x + k \) where \( k \) is an integer could be represented by **Option C**.