Question

find the value of $f(6)$.
$y = f(x)$
answer attempt 1 out of 2

Explanation:

Step1: Identify the vertex and intercepts

The graph of \( y = f(x) \) is a piecewise linear function. The vertex is at \( (2, 8) \) (since it's the peak), and it intersects the x - axis at \( x=-6 \) and \( x = 10 \), and the y - axis at \( y = 6 \).

Step2: Find the equation of the right - hand segment

The right - hand segment goes from \( (2, 8) \) to \( (10, 0) \). The slope \( m \) of a line passing through two points \( (x_1,y_1)=(2,8) \) and \( (x_2,y_2)=(10,0) \) is given by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
So, \( m=\frac{0 - 8}{10 - 2}=\frac{-8}{8}=- 1 \).
Using the point - slope form of a line \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(2,8) \) and \( m=-1 \), we get \( y-8=-1(x - 2) \), which simplifies to \( y=-x + 10 \).

Step3: Evaluate \( f(6) \)

We want to find \( f(6) \), and since \( 6\) is in the domain of the right - hand segment (because \( 2\leqslant6\leqslant10 \)), we substitute \( x = 6 \) into the equation \( y=-x + 10 \).
\( f(6)=-6 + 10=4 \).

Answer:

\( 4 \)