Question

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

Explanation:

Step1: Analyze the graph structure

The graph of \( y = f(x) \) is a piece - wise linear function. We can see that the right - hand part of the graph (after the peak) is a line segment. Let's first find the equation of the line for the part of the graph where \( x\geq2 \) (assuming the peak is at \( x = 2 \), \( y=8 \)) and \( x\leq10 \) (since it intersects the \( x \) - axis at \( x = 10 \), \( y = 0 \)).

The slope \( m \) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the points \((2,8)\) and \((10,0)\). Then \( m=\frac{0 - 8}{10 - 2}=\frac{- 8}{8}=- 1 \).

Step2: Use the point - slope form of a line

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((10,0)\) and \( m=-1 \), we have \( y-0=-1(x - 10) \), which simplifies to \( y=-x + 10 \).

Step3: Evaluate \( f(6) \)

We want to find \( f(6) \), so we substitute \( x = 6 \) into the equation \( y=-x + 10 \) (since \( 6 \) is between \( 2 \) and \( 10 \), it lies on this line segment).

Substitute \( x = 6 \) into \( y=-x + 10 \): \( y=-6 + 10=4 \).

Answer:

\( 4 \)