Pregunta
Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. \( f(x)=3 x^{2}+30 x-8 \) Does the quadratic function f have a minimum value or a maximum value? The function t has a maximum value. The function t has a minimum value. What is this minimum or maximum value? (Simplify your answer.)
Ask by Flynn Tran. in the United States
Mar 01,2025
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The quadratic function \( f(x) = 3x^{2} + 30x - 8 \) has a minimum value of -83.
Solución
For a quadratic function of the form f(x) = ax² + bx + c, the vertex occurs at x = -b/(2a). If a > 0, the parabola opens upward and the vertex represents a minimum value.
Here, a = 3 and b = 30, so the vertex is at:
x = -30/(2 × 3) = -30/6 = -5.
Now, substitute x = -5 into the function:
f(-5) = 3(-5)² + 30(-5) - 8
= 3(25) - 150 - 8
= 75 - 150 - 8
= -83.
Since a is positive, the quadratic function has a minimum value, and that minimum value is -83.
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Simplifique esta solución Beyond the Answer
The given quadratic function \( f(x) = 3x^2 + 30x - 8 \) has a minimum value because the coefficient of \( x^2 \) (which is 3) is positive. This indicates that the parabola opens upwards. To find the minimum value, we can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = 30 \). Plugging in these values gives: \[ x = -\frac{30}{2 \times 3} = -5. \] Now, substitute \( x = -5 \) back into the function to find the minimum value: \[ f(-5) = 3(-5)^2 + 30(-5) - 8 = 3(25) - 150 - 8 = 75 - 150 - 8 = -83. \] Thus, the function has a minimum value of -83.
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