Question

For a linear function, f(1) = 8 and f(7) = -10. If f(k) = 5, what is the value of k?

Answer 1

the basic form is y = mx + b

start by finding the slope of the line

\[m = \frac{f(7) - f(1)}{7 - 1}\]

or

\[m = \frac{-10 -8}{7 - 1}\]

so whats the slope of the line...?

Answer 2

oops... you need to find the equation before you can find k

Answer 3

Or you could solve the following for k instead of finding the equation of a line:
\[\frac{f(7)-f(1)}{7-1}=\frac{f(k)-f(1)}{k-1}\]

Answer 4

\(\bf f(1) = 8 \implies (1,8) \qquad f(7) = -10\implies (7,-10) \\ \quad \\ \quad \\
---------------------------\\
\begin{array}{lllll}
&x_1&y_1&x_2&y_2\\
&({\color{red}{ 1}}\quad ,&{\color{blue}{ 8}})\quad &({\color{red}{ 7}}\quad ,&{\color{blue}{ -10}})
\end{array}
\\\quad \\

slope = {\color{green}{ m}}= \cfrac{rise}{run} \implies \cfrac{{\color{blue}{ y_2}}-{\color{blue}{ y_1}}}{{\color{red}{ x_2}}-{\color{red}{ x_1}}}
\\ \quad \\

y-{\color{blue}{ y_1}}={\color{green}{ m}}(x-{\color{red}{ x_1}})\qquad \textit{plug in the values and solve for "y"}\\
\qquad \uparrow\\
\textit{point-slope form} \)

Answer 5

f(k) = 5 -> (k, 5) or "what is the value of "x" if y=5"?