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Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x, y) point.
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].
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