Question
A) What is the range of the quadratic function represented by the equation f(x) = − 5x
2 − 2?
(1 point)
y ≥ − 2
R
y ≥ − 5
y ≤ − 2
2 − 2?
(1 point)
y ≥ − 2
R
y ≥ − 5
y ≤ − 2
Answers
Answered by
GPT-4o mini
To find the range of the quadratic function \( f(x) = -5x^2 - 2 \), we start by identifying the characteristics of the function.
1. The coefficient of \( x^2 \) is negative (−5), indicating that the parabola opens downward.
2. The vertex of the parabola will give us the maximum value of the function.
We can find the vertex using the standard form \( f(x) = ax^2 + bx + c \). In this function, \( a = -5 \), \( b = 0 \), and \( c = -2 \). The x-coordinate of the vertex for a quadratic function is found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{0}{2 \cdot -5} = 0
\]
Now, we substitute \( x = 0 \) back into the function to find the y-coordinate of the vertex:
\[
f(0) = -5(0)^2 - 2 = -2
\]
Since the parabola opens downward, the y-coordinate of the vertex (\( -2 \)) is the maximum value of the function.
Therefore, the range of the function is all values of \( y \) that are less than or equal to the maximum value, which is:
\[
y \leq -2
\]
Thus, the correct answer is **y ≤ −2**.
1. The coefficient of \( x^2 \) is negative (−5), indicating that the parabola opens downward.
2. The vertex of the parabola will give us the maximum value of the function.
We can find the vertex using the standard form \( f(x) = ax^2 + bx + c \). In this function, \( a = -5 \), \( b = 0 \), and \( c = -2 \). The x-coordinate of the vertex for a quadratic function is found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{0}{2 \cdot -5} = 0
\]
Now, we substitute \( x = 0 \) back into the function to find the y-coordinate of the vertex:
\[
f(0) = -5(0)^2 - 2 = -2
\]
Since the parabola opens downward, the y-coordinate of the vertex (\( -2 \)) is the maximum value of the function.
Therefore, the range of the function is all values of \( y \) that are less than or equal to the maximum value, which is:
\[
y \leq -2
\]
Thus, the correct answer is **y ≤ −2**.