What is the range of this quadratic function?

y

=

x

2

+

3

x

–

10

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.

Learn how to find the range of the quadratic function y = x^2 + 3x - 10 with step-by-step guidance. Understand the vertex of a parabola, determine orientation, and apply the vertex formula. Suitable for high school students studying quadratic functions.

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