Question

question
determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.
$f(x) = (x - 9)(x - 5)$
answer attempt 1 out of 2
the
value is
.
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Explanation:

Step1: Expand the function

First, we expand the quadratic function \( f(x)=(x - 9)(x - 5) \). Using the FOIL method: \( (x-9)(x - 5)=x^2-5x-9x + 45=x^2-14x + 45 \).

Step2: Determine the direction of the parabola

For a quadratic function in the form \( f(x)=ax^2+bx + c \), the coefficient \( a \) determines if the parabola opens up or down. Here, \( a = 1 \) (since the coefficient of \( x^2 \) is 1). Since \( a>0 \), the parabola opens upward, so the function has a minimum value.

Step3: Find the x-coordinate of the vertex

The x-coordinate of the vertex of a quadratic function \( f(x)=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). For \( f(x)=x^2-14x + 45 \), \( a = 1 \) and \( b=-14 \). So, \( x=-\frac{-14}{2\times1}=\frac{14}{2}=7 \).

Step4: Find the minimum value

Now, we substitute \( x = 7 \) into the function \( f(x)=(x - 9)(x - 5) \) (we can also use the expanded form). Using the factored form: \( f(7)=(7 - 9)(7 - 5)=(-2)(2)=-4 \).

Answer:

The minimum value is \(-4\).