To identify the key characteristics of the quadratic function \( y = x^2 - 3x - 4 \), we can analyze the function in several ways:
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Standard Form: The function is in standard form \( y = ax^2 + bx + c \), where:
- \( a = 1 \) (the coefficient of \( x^2 \))
- \( b = -3 \)
- \( c = -4 \)
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Direction of the Parabola: Since \( a > 0 \), the parabola opens upwards.
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Vertex: The vertex of the quadratic function can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting in \( b = -3 \) and \( a = 1 \): \[ x = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \] To find the corresponding \( y \)-coordinate of the vertex, substitute \( x = \frac{3}{2} \) back into the original equation: \[ y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) - 4 = \frac{9}{4} - \frac{9}{2} - 4 = \frac{9}{4} - \frac{18}{4} - \frac{16}{4} = \frac{9 - 18 - 16}{4} = \frac{-25}{4} \] Therefore, the vertex is at: \[ \left(\frac{3}{2}, -\frac{25}{4}\right) \]
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Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, which is given by: \[ x = \frac{3}{2} \]
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Y-intercept: The y-intercept occurs when \( x = 0 \): \[ y(0) = 0^2 - 3(0) - 4 = -4 \] So, the y-intercept is at \( (0, -4) \).
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X-intercepts (Roots): The x-intercepts can be found by setting \( y = 0 \): \[ 0 = x^2 - 3x - 4 \] Factoring the quadratic yields: \[ (x - 4)(x + 1) = 0 \] Thus, the x-intercepts are: \[ x = 4 \quad \text{and} \quad x = -1 \] So, the x-intercepts are \( (4, 0) \) and \( (-1, 0) \).
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Opening and Range: Since the parabola opens upwards, the minimum value of the function (the y-coordinate of the vertex) is \( -\frac{25}{4} \). Therefore, the range of the function is: \[ y \geq -\frac{25}{4} \]
Summary of Key Characteristics:
- Direction: Opens upwards
- Vertex: \( \left( \frac{3}{2}, -\frac{25}{4} \right) \)
- Axis of Symmetry: \( x = \frac{3}{2} \)
- Y-intercept: \( (0, -4) \)
- X-intercepts: \( (4, 0) \) and \( (-1, 0) \)
- Range: \( y \geq -\frac{25}{4} \)