Question

The axis of symmetry for the graph of the function f start bracket x end bracket equals one-quarter x square plus b x plus 10 is x=6. What is the value of b?

−12
−3

3

Answer 1

The axis of symmetry for a quadratic function given by the standard form \( f(x) = ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

In your function \( f(x) = \frac{1}{4}x^2 + bx + 10 \), the coefficient \( a \) is \( \frac{1}{4} \).

Given that the axis of symmetry is \( x = 6 \), we can set up the equation:

\[
6 = -\frac{b}{2 \cdot \frac{1}{4}}
\]

Calculating the denominator:

\[
2 \cdot \frac{1}{4} = \frac{1}{2}
\]

Now substituting this into the equation:

\[
6 = -\frac{b}{\frac{1}{2}} = -2b
\]

Now we can solve for \( b \):

\[
-2b = 6
\]

Dividing both sides by -2:

\[
b = -3
\]

Thus, the value of \( b \) is \(\boxed{-3}\).