To find the vertex of the quadratic equation [tex]\( y = x^2 + 4x - 5 \)[/tex], we follow these steps:
1. Identify the coefficients:
- The quadratic equation is given in the form [tex]\( y = ax^2 + bx + c \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex].
2. Calculate the x-coordinate ([tex]\( h \)[/tex]) of the vertex:
- The formula to find the x-coordinate of the vertex [tex]\( h \)[/tex] is:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- So,
[tex]\[
h = -\frac{4}{2 \times 1} = -\frac{4}{2} = -2
\][/tex]
- Thus, [tex]\( h = -2 \)[/tex].
4. Calculate the y-coordinate ([tex]\( k \)[/tex]) of the vertex:
- The formula for the y-coordinate of the vertex [tex]\( k \)[/tex] is:
[tex]\[
k = c - \frac{b^2 - 4ac}{4a}
\][/tex]
5. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -5 \)[/tex]
- So,
[tex]\[
k = -5 - \frac{4^2 - 4 \times 1 \times -5}{4 \times 1}
\][/tex]
- Calculate the numerator inside the fraction:
[tex]\[
4^2 - 4 \times 1 \times -5 = 16 + 20 = 36
\][/tex]
- Now substitute this back into the formula:
[tex]\[
k = -5 - \frac{36}{4 \times 1} = -5 - 9 = -14
\][/tex]
- Thus, [tex]\( k = -14 \)[/tex].
6. Write the vertex as an ordered pair:
- The vertex [tex]\( (h, k) \)[/tex] is:
[tex]\[
(-2, -14)
\][/tex]
Therefore, the vertex of the graph of the equation [tex]\( y = x^2 + 4x - 5 \)[/tex] is [tex]\( (-2, -14) \)[/tex].
