To find the equation of the axis of symmetry for the parabola represented by [tex]\( y = -x^2 + 4x + 5 \)[/tex] and determine the maximum value of the expression, follow these steps:
1. Identify the coefficients:
The given quadratic equation is [tex]\( y = -x^2 + 4x + 5 \)[/tex]. Here, the coefficients are:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
2. Calculate the axis of symmetry:
The axis of symmetry for a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] is given by the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into this formula, we get:
[tex]\[
x = -\frac{4}{2(-1)} = -\frac{4}{-2} = 2
\][/tex]
Thus, the equation of the axis of symmetry is [tex]\( x = 2 \)[/tex].
3. Determine the maximum value:
Since the parabola opens downwards (i.e., [tex]\( a < 0 \)[/tex]), the vertex represents the maximum point. To find the maximum value, substitute [tex]\( x = 2 \)[/tex] (the axis of symmetry) back into the original equation:
[tex]\[
y = -x^2 + 4x + 5
\][/tex]
[tex]\[
y = -(2)^2 + 4(2) + 5
\][/tex]
[tex]\[
y = -4 + 8 + 5
\][/tex]
[tex]\[
y = 9
\][/tex]
Therefore, the maximum value of the expression [tex]\( -x^2 + 4x + 5 \)[/tex] is 9.
Conclusion:
- The equation of the axis of symmetry is [tex]\( x = 2 \)[/tex].
- The maximum value of the expression [tex]\( -x^2 + 4x + 5 \)[/tex] is [tex]\( 9 \)[/tex].
