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 \( x = -1.25 \).