Question

Graph the quadratic y=2x^2+5x−10. What is the vertex of this quadratic? (round to the nearest hundredth if needed)

Answer 1

To find the vertex of the quadratic equation \( y = 2x^2 + 5x - 10 \), we can use the formula for the vertex of a parabola given by the equation \( y = ax^2 + bx + c \). The x-coordinate of the vertex can be found using the formula:

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

Here, \( a = 2 \) and \( b = 5 \).

1. **Calculate the x-coordinate of the vertex**:

\[
x = -\frac{5}{2 \cdot 2} = -\frac{5}{4} = -1.25
\]

2. **Calculate the y-coordinate of the vertex** by substituting \( x = -1.25 \) back into the quadratic equation:

\[
y = 2(-1.25)^2 + 5(-1.25) - 10
\]

Calculating \( (-1.25)^2 \):

\[
(-1.25)^2 = 1.5625
\]

Substituting this value back into the equation:

\[
y = 2(1.5625) + 5(-1.25) - 10
\]

Now calculate:

\[
y = 3.125 - 6.25 - 10
\]
\[
y = 3.125 - 16.25 = -13.125
\]

Thus, the vertex of the quadratic \( y = 2x^2 + 5x - 10 \) is:

\[
(-1.25, -13.125)
\]

In rounded form (to the nearest hundredth, if needed):

- Vertex: **(-1.25, -13.13)**

### Graphing the Quadratic

To graph the quadratic, we can plot the vertex and a few additional points. Here's a simple way to do so:

1. **Vertex**: \( (-1.25, -13.13) \)
2. **Choose x-values** around the vertex, for example, \( x = -3, -2, -1, 0, 1 \).

Calculating y for these:

- \( x = -3 \):
\[
y = 2(-3)^2 + 5(-3) - 10 = 18 - 15 - 10 = -7
\]

- \( x = -2 \):
\[
y = 2(-2)^2 + 5(-2) - 10 = 8 - 10 - 10 = -12
\]

- \( x = -1 \):
\[
y = 2(-1)^2 + 5(-1) - 10 = 2 - 5 - 10 = -13
\]

- \( x = 0 \):
\[
y = 2(0)^2 + 5(0) - 10 = -10
\]

- \( x = 1 \):
\[
y = 2(1)^2 + 5(1) - 10 = 2 + 5 - 10 = -3
\]

Now we can plot the points:

- \( (-3, -7) \)
- \( (-2, -12) \)
- \( (-1, -13) \)
- \( (0, -10) \)
- \( (1, -3) \)

This will help us sketch the parabola.

The parabola opens upwards (since \( a = 2 > 0 \)), and the graph will be symmetric about the vertical line through the vertex.

### Summary
- Vertex: \((-1.25, -13.13)\)