Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.

f, of, x, equals, 3, left parenthesis, x, minus, 4, right parenthesis, squared, minus, 7

f(x)=

3(x−4) 

2

 −7

The function given is:

\[
f(x) = 3(x - 4)^2 - 7
\]

This is a quadratic function in vertex form: \( f(x) = a(x - h)^2 + k \), where \( a \), \( h \), and \( k \) are constants. For this function:

- \( a = 3 \)
- \( h = 4 \)
- \( k = -7 \)

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

Since \( a = 3 \) (a positive number), the parabola opens upwards. This means the function has a **minimum** value.

### Step 2: Find the minimum value of the function

For a function in the form \( f(x) = a(x - h)^2 + k \), the minimum (or maximum) value occurs at \( x = h \). Here, \( h = 4 \). 

Now substitute \( x = 4 \) into the function to find the minimum value:

\[
f(4) = 3(4 - 4)^2 - 7 = 3 \cdot 0 - 7 = -7
\]

### Conclusion

The function has a **minimum** value of \(-7\), which occurs at \( x = 4 \).

Would you like more details or have any questions?

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1. How do you determine whether a quadratic function has a minimum or maximum value?
2. Why does the sign of \( a \) in \( f(x) = ax^2 + bx + c \) affect the parabola's direction?
3. How do we find the vertex of a quadratic function in standard form?
4. How can we rewrite quadratic functions in vertex form from standard form?
5. What are real-life examples where minimum or maximum values of a quadratic function are useful?

**Tip:** The vertex form of a quadratic function makes it easy to identify the maximum or minimum value directly.

Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.

f(x) = 3(x - 4)^2 - 7

Learn how to determine whether a quadratic function has a minimum or maximum value and find that value. This guide walks through analyzing f(x) = 3(x - 4)^2 - 7 using vertex form and algebraic concepts, ideal for Grades 9-10 students.

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