To identify the graph of the quadratic function \( g(x) = 4x^2 + 24x + 38 \), we can start by analyzing its key characteristics.
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Form of the Function: The given function is in the standard form \( g(x) = ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 24 \)
- \( c = 38 \)
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Direction of the Parabola: Since \( a = 4 \) is positive, the parabola opens upward.
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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 \( (-3, 2) \).
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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 \( (0, 38) \).
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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.
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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 \( ( -3, 2) \).