Question

Suppose you were to graph a parabola given by the equation Y=1/4x^2+3x-1. What orders pair would be the given vertex of the graph

Answer 1

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 in the standard form \( Y = ax^2 + bx + c \). The x-coordinate of the vertex is given by the formula:

\[
x = -\frac{b}{2a}
\]

For the equation \( Y = \frac{1}{4}x^2 + 3x - 1 \), the coefficients are:
- \( a = \frac{1}{4} \)
- \( b = 3 \)

Substituting these values into the vertex formula:

\[
x = -\frac{3}{2 \cdot \frac{1}{4}} = -\frac{3}{\frac{1}{2}} = -3 \times 2 = -6
\]

Now that we have the x-coordinate of the vertex, we can find the y-coordinate 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 at the point \( (-6, -10) \).

So, the ordered pair that represents the vertex of the graph is:

\[
\boxed{(-6, -10)}
\]