To find the coordinates of the vertex of the parabola given by the equation [tex]\( y = 5x^2 + 20x + 35 \)[/tex], we need to use the vertex formula for a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex].
1. Identify constants a, b, and c:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 20 \)[/tex]
- [tex]\( c = 35 \)[/tex]
2. Calculate the x-coordinate of the vertex:
The x-coordinate of the vertex for a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x = -\frac{20}{2 \cdot 5} = -\frac{20}{10} = -2
\][/tex]
So, the x-coordinate of the vertex is [tex]\( x = -2 \)[/tex].
3. Calculate the y-coordinate of the vertex:
To find the y-coordinate, substitute [tex]\( x = -2 \)[/tex] back into the original equation:
[tex]\[
y = 5(-2)^2 + 20(-2) + 35
\][/tex]
Calculate each term step-by-step:
- [tex]\( 5(-2)^2 = 5 \cdot 4 = 20 \)[/tex]
- [tex]\( 20(-2) = -40 \)[/tex]
- The constant term [tex]\( c = 35 \)[/tex]
Now, sum these values:
[tex]\[
y = 20 - 40 + 35
\][/tex]
[tex]\[
y = 20 - 40 = -20
\][/tex]
[tex]\[
y = -20 + 35 = 15
\][/tex]
So, the y-coordinate of the vertex is [tex]\( y = 15 \)[/tex].
4. Combine the x and y coordinates:
Therefore, the coordinates of the vertex are:
[tex]\[
(-2, 15)
\][/tex]
So, the vertex of the parabola [tex]\( y = 5x^2 + 20x + 35 \)[/tex] is at the point [tex]\( (-2, 15) \)[/tex].
