Question
What is the range of this quadratic function?
y=x2+4x+4
choices
{y|y ≤ 0}
{y|y ≥0}
{y|y ≥-2}
all real numbers
Answers
GPT-4o mini
To determine the range of the quadratic function \( y = x^2 + 4x + 4 \), we can first rewrite it in vertex form.
1. **Complete the square:**
\[
y = x^2 + 4x + 4 = (x + 2)^2
\]
2. **Identify the vertex:**
The function \( (x + 2)^2 \) reaches its minimum when \( x + 2 = 0 \), which occurs at \( x = -2 \). Plugging this back into the function gives:
\[
y = (0)^2 = 0.
\]
So, the vertex of the parabola is at the point \( (-2, 0) \) and since it opens upwards (as the coefficient of \( x^2 \) is positive), the minimum value of \( y \) is 0.
3. **Determine the range:**
Since the function opens upwards and has a minimum value of \( 0 \), the range of \( y \) is all values of \( y \) greater than or equal to 0.
Therefore, the range of the function is:
\[
\{y | y \geq 0\}
\]
So, the correct choice is:
\(\{y | y \geq 0\}\).
1. **Complete the square:**
\[
y = x^2 + 4x + 4 = (x + 2)^2
\]
2. **Identify the vertex:**
The function \( (x + 2)^2 \) reaches its minimum when \( x + 2 = 0 \), which occurs at \( x = -2 \). Plugging this back into the function gives:
\[
y = (0)^2 = 0.
\]
So, the vertex of the parabola is at the point \( (-2, 0) \) and since it opens upwards (as the coefficient of \( x^2 \) is positive), the minimum value of \( y \) is 0.
3. **Determine the range:**
Since the function opens upwards and has a minimum value of \( 0 \), the range of \( y \) is all values of \( y \) greater than or equal to 0.
Therefore, the range of the function is:
\[
\{y | y \geq 0\}
\]
So, the correct choice is:
\(\{y | y \geq 0\}\).