Question

Graph the quadratic y=2x2+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 function \( y = 2x^2 + 5x - 10 \), we can use the formula for the x-coordinate of the vertex of a parabola given by the standard form \( y = ax^2 + bx + c \):

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

In this case, the coefficients are:
- \( a = 2 \)
- \( b = 5 \)

Substituting these values into the formula gives:

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

Next, we find 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 \( (-1.25)^2 \):

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

Now substituting this value in:

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

Calculating \( 2(1.5625) \):

\[
2(1.5625) = 3.125
\]

Calculating \( 5(-1.25) \):

\[
5(-1.25) = -6.25
\]

Now combine the values:

\[
y = 3.125 - 6.25 - 10
\]

Combining the constants:

\[
y = 3.125 - 6.25 = -3.125
\]

Then:

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

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

\[
\left(-1.25, -13.125\right)
\]

So, to round the vertex to the nearest hundredth, we can write it as:

\[
\text{Vertex: } (-1.25, -13.13)
\]

In summary:

- The vertex of the quadratic \( y = 2x^2 + 5x - 10 \) is \( (-1.25, -13.13) \).

To graph the function, plot the vertex and some other points by choosing values for \( x \) near the vertex, or use the standard parabolic shape features (opening upwards because \( a > 0 \)). The graph is symmetrical about the line \( x = -1.25 \).