To find the range of the quadratic function $y = x^2 + 3x - 10$, we need to determine the minimum or maximum value of the function, depending on its orientation, and then analyze the behavior of the function as $x$ tends to infinity or negative infinity.

Step 1: Identify the orientation of the parabola

The quadratic function $y = ax^2 + bx + c$ opens upwards if $a > 0$ and downwards if $a < 0$. In this case, $a = 1$ (since the coefficient of $x^2$ is 1), so the parabola opens upwards.

Step 2: Find the vertex

The vertex of the parabola is the point where the function reaches its minimum value (since it opens upwards). The $x$-coordinate of the vertex can be found using the formula:

$x = -\frac{b}{2a}$

Here, $a = 1$ and $b = 3$. Substituting these values:

$x = -\frac{3}{2(1)} = -\frac{3}{2}$

Now, substitute this $x$-value back into the original function to find the corresponding $y$-value:

$y = \left(-\frac{3}{2}\right)^2 + 3\left(-\frac{3}{2}\right) - 10$

$y = \frac{9}{4} - \frac{9}{2} - 10$

$y = \frac{9}{4} - \frac{18}{4} - \frac{40}{4} = \frac{9 - 18 - 40}{4} = \frac{-49}{4}$

So, the vertex of the parabola is at $\left(-\frac{3}{2}, -\frac{49}{4}\right)$.

Step 3: Determine the range

Since the parabola opens upwards and the minimum value of $y$ is $-\frac{49}{4}$, the range of the function is:

$\text{Range} = \left[-\frac{49}{4}, \infty\right)$

Final Answer:

The range of the quadratic function $y = x^2 + 3x - 10$ is $\left[-\frac{49}{4}, \infty\right)$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you determine the axis of symmetry for a quadratic function?
2. Can the range of a quadratic function ever be all real numbers?
3. How would the range change if the parabola opened downwards?
4. What is the significance of the discriminant in a quadratic equation?
5. How do you find the y-intercept of a quadratic function?

Tip: The vertex form of a quadratic equation can be very useful for quickly identifying the vertex and range of the function.