y=x^2+4x+9 is defined over the restriced interval, [-6,1]. what is the corresponding range?

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).

y = x^2 + 4x + 9 is defined over the restricted interval [-6, 1]. What is the corresponding range?

Discover how to find the range of the quadratic function y = x^2 + 4x + 9 over the restricted interval [-6, 1]. This solution walks through the process of finding the vertex and evaluating endpoints, covering key algebraic concepts such as the vertex formula. Suitable for students in Grades 9-12.

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