Question

Identify the graph of g of x equals 4x squared + 24x + 38

Answer 1

To identify the graph of the quadratic function \( g(x) = 4x^2 + 24x + 38 \), we can start by analyzing its key characteristics.

1. **Form of the Function**: The given function is in the standard form \( g(x) = ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 24 \)
- \( c = 38 \)

2. **Direction of the Parabola**: Since \( a = 4 \) is positive, the parabola opens upward.

3. **Vertex**: The vertex \( x \)-coordinate can be found using the formula:
\[
x = -\frac{b}{2a} = -\frac{24}{2 \times 4} = -\frac{24}{8} = -3
\]
To find the \( y \)-coordinate of the vertex, substitute \( x = -3 \) back into the function:
\[
g(-3) = 4(-3)^2 + 24(-3) + 38 = 4(9) - 72 + 38 = 36 - 72 + 38 = 2
\]
Therefore, the vertex is at \( (-3, 2) \).

4. **Y-Intercept**: To find the \( y \)-intercept, set \( x = 0 \):
\[
g(0) = 4(0)^2 + 24(0) + 38 = 38
\]
Thus, the \( y \)-intercept is at \( (0, 38) \).

5. **X-Intercepts**: To find the \( x \)-intercepts, set \( g(x) = 0 \):
\[
4x^2 + 24x + 38 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
x = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 4 \cdot 38}}{2 \cdot 4} = \frac{-24 \pm \sqrt{576 - 608}}{8} = \frac{-24 \pm \sqrt{-32}}{8}
\]
Since the discriminant is negative, there are no real \( x \)-intercepts. The function does not cross the \( x \)-axis.

6. **Graph Shape**: The parabola will be symmetrical about the vertical line \( x = -3 \), and the vertex \( (-3, 2) \) will be the minimum point.

In summary, the graph of \( g(x) = 4x^2 + 24x + 38 \) is a parabola that opens upwards, has a vertex at \( (-3, 2) \), a \( y \)-intercept at \( (0, 38) \), and no \( x \)-intercepts. The overall shape will be a "U" shape, positioned above the x-axis, touching the vertex at \( ( -3, 2) \).