find the value of $f(6)$.\n$y = f(x)$\nanswer attempt 1 out of 2
find the value of $f(6)$.\n$y = f(x)$\nanswer attempt 1 out of 2
Answer
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 )