The function given in the image is $f(x) = -3x^2 + 6x - 8$.

Part (a)

To determine if the function has a minimum or maximum value, we look at the coefficient of $x^2$ in the quadratic function $f(x) = ax^2 + bx + c$.

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

In this case, $a = -3$, which is less than 0, so the function has a maximum value.

Part (b)

To find the maximum value and where it occurs, we use the vertex formula: $x = \frac{-b}{2a}$ Here, $a = -3$ and $b = 6$. Substituting these values into the formula: $x = \frac{-6}{2(-3)} = 1$

Now substitute $x = 1$ back into the function $f(x)$ to find the maximum value: $f(1) = -3(1)^2 + 6(1) - 8 = -3 + 6 - 8 = -5$ So, the maximum value is $-5$, and it occurs at $x = 1$.

Part (c)

• The domain of any quadratic function is all real numbers: $(-\infty, \infty)$.
• The range of the function depends on the maximum value. Since the maximum value is $-5$, the range is $(-\infty, -5]$.

Summary:

• (a) The function has a maximum value.
• (b) The maximum value is $-5$, and it occurs at $x = 1$.
• (c) The domain is $(-\infty, \infty)$, and the range is $(-\infty, -5]$.

Would you like further details on any part of the solution?

Here are 5 related questions:

1. How do you determine the vertex of a quadratic function?
2. What does the sign of the leading coefficient in a quadratic equation tell you about its graph?
3. Can a quadratic function have both a minimum and a maximum value?
4. How do you find the axis of symmetry for a quadratic function?
5. What is the general form of a quadratic equation?

Tip: Always check the sign of the $x^2$ coefficient to know if a quadratic function has a minimum or maximum.