Step 1: Determine if the quadratic function has a minimum or maximum value

The given quadratic function is:
$f(x) = 3x^2 + 12x - 8$

For a quadratic function of the form $f(x) = ax^2 + bx + c$:

• If $a > 0$, the parabola opens upward, and the function has a minimum value.
• If $a < 0$, the parabola opens downward, and the function has a maximum value.

Here, $a = 3 > 0$, so the function has a minimum value.

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Step 2: Find the minimum value

The minimum (or maximum) value of a quadratic function occurs at the vertex.
For a quadratic function $f(x) = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by: $x = -\frac{b}{2a}$ For our function $f(x) = 3x^2 + 12x - 8$, we have:

• $a = 3$
• $b = 12$

Now, substitute into the vertex formula: $x = -\frac{12}{2(3)} = -\frac{12}{6} = -2$

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Step 3: Find the function value at the vertex

Now that we know the $x$-coordinate of the vertex is $x = -2$, substitute it into the function to find the corresponding $f(x)$ value: $f(-2) = 3(-2)^2 + 12(-2) - 8$ $f(-2) = 3(4) + 12(-2) - 8 = 12 - 24 - 8 = -20$

So, the minimum value of the function is: $f(-2) = -20$

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Final Answer

1. The function has a minimum value because the coefficient of $x^2$ is positive.
2. The minimum value is $-20$.

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Would you like further details or another example?

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Here are 5 related questions you might consider:

1. How do you determine if a quadratic function opens up or down?
2. What is the vertex form of a quadratic function?
3. How does completing the square help in finding the vertex?
4. How do you find the axis of symmetry for a quadratic function?
5. What happens to the graph if the value of $a$ is increased or decreased?

Tip: The vertex gives the turning point of a parabola, which is either its minimum or maximum depending on the coefficient of $x^2$.