To find the equation of the axis of symmetry for the parabola given by the equation [tex]\( y = -x^2 + 2x + 3 \)[/tex], you can use the standard formula for the axis of symmetry of a parabola of the form [tex]\( y = ax^2 + bx + c \)[/tex], which is [tex]\( x = -\frac{b}{2a} \)[/tex].
Here are the steps:
1. Identify the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] from the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex].
- For the equation [tex]\( y = -x^2 + 2x + 3 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
2. Substitute these values into the formula [tex]\( x = -\frac{b}{2a} \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- So, the formula becomes [tex]\( x = -\frac{2}{2 \cdot (-1)} \)[/tex].
3. Simplify the expression:
- [tex]\( x = -\frac{2}{-2} \)[/tex]
- [tex]\( x = 1 \)[/tex]
Therefore, the equation of the axis of symmetry for the given parabola [tex]\( y = -x^2 + 2x + 3 \)[/tex] is:
[tex]\[ x = 1 \][/tex]
This means that the axis of symmetry is a vertical line that passes through the point [tex]\( x = 1 \)[/tex] on the x-axis.
