To find the minimum or maximum value of the quadratic function:

$f(x) = 2x^2 - 16x + 30$

we use the vertex formula. The vertex of a quadratic function $ax^2 + bx + c$ occurs at:

$x = \frac{-b}{2a}$

Step 1: Identify coefficients

• $a = 2$
• $b = -16$
• $c = 30$

Step 2: Compute the vertex $x$-coordinate

$x = \frac{-(-16)}{2(2)} = \frac{16}{4} = 4$

Step 3: Find the function value at $x = 4$

$f(4) = 2(4)^2 - 16(4) + 30$

$= 2(16) - 64 + 30$

$= 32 - 64 + 30 = -2$

Step 4: Determine if it's a minimum or maximum

Since the coefficient of $x^2$ (which is $2$) is positive, the parabola opens upward, meaning the vertex represents a minimum value.

Thus, the minimum value is:

$\mathbf{-2}$

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Would you like a detailed graph or further explanations?

Here are five related questions:

1. How do you determine if a quadratic function has a minimum or maximum?
2. How do you complete the square to find the vertex of a quadratic function?
3. What are the real roots of $f(x) = 2x^2 - 16x + 30$, if any?
4. How do you find the axis of symmetry of a quadratic function?
5. How do transformations affect the graph of a quadratic function?

Tip:

For any quadratic function $ax^2 + bx + c$, if $a > 0$, the function has a minimum at the vertex. If $a < 0$, it has a maximum at the vertex. 🚀