Question
What is the range of this quadratic function?
y=x2–10x+21
choices
{y|y ≤ 5}
{y|y ≥5}
{y|y ≥-4}
all real numbers
y=x2–10x+21
choices
{y|y ≤ 5}
{y|y ≥5}
{y|y ≥-4}
all real numbers
Answers
Answered by
GPT-4o mini
To find the range of the quadratic function \( y = x^2 - 10x + 21 \), we first need to rewrite it in vertex form by completing the square.
1. Start with the original equation:
\[
y = x^2 - 10x + 21
\]
2. To complete the square, take half of the coefficient of \( x \) (which is -10), square it, and add and subtract that value inside the function:
\[
\text{Half of -10 is -5, and } (-5)^2 = 25.
\]
So, we can rewrite the equation as:
\[
y = (x^2 - 10x + 25) - 25 + 21
\]
\[
y = (x - 5)^2 - 4
\]
3. Now, we can identify the vertex of the parabola. The vertex form \( y = a(x - h)^2 + k \) shows that the vertex is at \( (h, k) \), where \( h = 5 \) and \( k = -4 \).
Thus, the vertex is \( (5, -4) \).
4. Since the coefficient of \( (x - 5)^2 \) is positive, the parabola opens upwards. This means that the minimum value of \( y \) occurs at the vertex.
5. Therefore, the minimum value of the function is \( -4 \) at \( x = 5\). Hence, the function can take any value equal to or greater than \( -4 \).
6. Thus, the range of the function is:
\[
y \geq -4
\]
Looking at the choices provided, the correct answer is:
\[
\{y | y \geq -4\}
\]
1. Start with the original equation:
\[
y = x^2 - 10x + 21
\]
2. To complete the square, take half of the coefficient of \( x \) (which is -10), square it, and add and subtract that value inside the function:
\[
\text{Half of -10 is -5, and } (-5)^2 = 25.
\]
So, we can rewrite the equation as:
\[
y = (x^2 - 10x + 25) - 25 + 21
\]
\[
y = (x - 5)^2 - 4
\]
3. Now, we can identify the vertex of the parabola. The vertex form \( y = a(x - h)^2 + k \) shows that the vertex is at \( (h, k) \), where \( h = 5 \) and \( k = -4 \).
Thus, the vertex is \( (5, -4) \).
4. Since the coefficient of \( (x - 5)^2 \) is positive, the parabola opens upwards. This means that the minimum value of \( y \) occurs at the vertex.
5. Therefore, the minimum value of the function is \( -4 \) at \( x = 5\). Hence, the function can take any value equal to or greater than \( -4 \).
6. Thus, the range of the function is:
\[
y \geq -4
\]
Looking at the choices provided, the correct answer is:
\[
\{y | y \geq -4\}
\]
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