To find the range of the function $y = x^2 + 4x + 9$ over the restricted interval $[-6, 1]$, we need to follow these steps:

1. Find the vertex:

The function $y = x^2 + 4x + 9$ is a quadratic equation, and its graph is a parabola. Since the coefficient of $x^2$ is positive, the parabola opens upwards, and the minimum point of the parabola occurs at its vertex.

The x-coordinate of the vertex is given by the formula: $x = \frac{-b}{2a}$ where $a = 1$ and $b = 4$. Substituting these values: $x = \frac{-4}{2(1)} = -2$

2. Evaluate the function at the vertex:

Now substitute $x = -2$ into the function to find the corresponding y-coordinate (the minimum value): $y = (-2)^2 + 4(-2) + 9 = 4 - 8 + 9 = 5$ Thus, the minimum value of the function is $y = 5$.

3. Evaluate the function at the endpoints:

Next, evaluate the function at the endpoints of the interval $[-6, 1]$.

• At $x = -6$: $y = (-6)^2 + 4(-6) + 9 = 36 - 24 + 9 = 21$
• At $x = 1$: $y = (1)^2 + 4(1) + 9 = 1 + 4 + 9 = 14$

4. Determine the range:

• The minimum value is $y = 5$ (at $x = -2$),
• The maximum value is $y = 21$ (at $x = -6$).

Thus, the range of the function over the interval $[-6, 1]$ is: $[5, 21]$

Summary:

The range of the function $y = x^2 + 4x + 9$ over the interval $[-6, 1]$ is $[5, 21]$.

Do you need further details or have any questions?

Related Questions:

1. What is the domain and range of a general quadratic function?
2. How can you find the vertex of a quadratic function using its coefficients?
3. How does the shape of a parabola change with different coefficients?
4. How would the range change if the interval was $[-7, 2]$?
5. Can the quadratic formula be used to find the range of a quadratic function?

Tip:

Always remember that for a quadratic function, the vertex gives the minimum or maximum value, depending on the direction of the parabola (upward or downward).