Question

Which statements are true about the function y=x^2+2x-3
A.
The graph of the function opens up.

B.
The graph of the function opens down.

C.
The graph of the function has a vertex at (1,0)
.

D.
The graph of the function has a vertex at (-1,-4)
.
The graph of the function has an axis of symmetry of x=0
.

F.
The graph of the function has an axis of symmetry of x=-1

Answer 1

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.