Question
suppose you were to graph a parabola given by the equation y = 1/4x^2 + 3x - 1. which ordered pair would be the vertex of the graph
Answers
GPT-4o mini
To find the vertex of the parabola given by the equation \( y = \frac{1}{4}x^2 + 3x - 1 \), we can use the vertex formula. For a quadratic equation of the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = \frac{1}{4} \) and \( b = 3 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{3}{2 \cdot \frac{1}{4}} = -\frac{3}{\frac{1}{2}} = -3 \cdot 2 = -6
\]
Next, we can find the y-coordinate of the vertex by substituting \( x = -6 \) back into the original equation:
\[
y = \frac{1}{4}(-6)^2 + 3(-6) - 1
\]
\[
y = \frac{1}{4}(36) - 18 - 1
\]
\[
y = 9 - 18 - 1
\]
\[
y = 9 - 19 = -10
\]
Thus, the vertex of the parabola is:
\[
\text{Vertex} = (-6, -10)
\]
The ordered pair that represents the vertex of the graph is \((-6, -10)\).
\[
x = -\frac{b}{2a}
\]
In this case, \( a = \frac{1}{4} \) and \( b = 3 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{3}{2 \cdot \frac{1}{4}} = -\frac{3}{\frac{1}{2}} = -3 \cdot 2 = -6
\]
Next, we can find the y-coordinate of the vertex by substituting \( x = -6 \) back into the original equation:
\[
y = \frac{1}{4}(-6)^2 + 3(-6) - 1
\]
\[
y = \frac{1}{4}(36) - 18 - 1
\]
\[
y = 9 - 18 - 1
\]
\[
y = 9 - 19 = -10
\]
Thus, the vertex of the parabola is:
\[
\text{Vertex} = (-6, -10)
\]
The ordered pair that represents the vertex of the graph is \((-6, -10)\).