To evaluate the statements about the quadratic function \( y = x^2 + 2x - 3 \), let's analyze the properties of the function.
A. The graph of the function opens up.
True. The coefficient of \( x^2 \) is positive (1), so the parabola opens upwards.
B. The graph of the function opens down.
False. As mentioned above, the parabola opens upwards.
C. The graph of the function has a vertex at (1,0).
False. To find the vertex, we use the vertex formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 2 \): \[ x = -\frac{2}{2 \cdot 1} = -1 \] We can find the y-coordinate of the vertex by substituting \( x = -1 \) back into the function: \[ y = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4 \] Thus, the vertex is at \((-1, -4)\).
D. The graph of the function has a vertex at (-1,-4).
True. As calculated above, the vertex is indeed at \((-1, -4)\).
E. The graph of the function has an axis of symmetry of x=0.
False. The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is at \( x = -1 \), the axis of symmetry is \( x = -1 \).
F. The graph of the function has an axis of symmetry of x=-1.
True. As derived, the axis of symmetry is \( x = -1 \).
Summary of True Statements
- A. True
- B. False
- C. False
- D. True
- E. False
- F. True
Thus, the true statements are A, D, and F.