Question

consider the following.
f(1) = 2, f(0) = 9
(a) write the linear function f that has the given function values.
f(x) =
(b) sketch the graph of the function.

Explanation:

Step1: Recall linear - function form

A linear function is of the form $f(x)=mx + b$, where $m$ is the slope and $b$ is the y - intercept.
Given $f(0)=9$, substituting $x = 0$ into $f(x)=mx + b$ gives $f(0)=m\times0 + b=b$. So, $b = 9$.

Step2: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.
We have the points $(0,9)$ (since $f(0)=9$) and $(1,2)$ (since $f(1)=2$).
So, $m=\frac{2 - 9}{1 - 0}=\frac{-7}{1}=-7$.

Step3: Write the linear function

Substituting $m=-7$ and $b = 9$ into $f(x)=mx + b$, we get $f(x)=-7x + 9$.

Answer:

$f(x)=-7x + 9$

For part (b), to sketch the graph of $y=-7x + 9$:

• The y - intercept is 9. So, the line crosses the y - axis at the point $(0,9)$.
• The slope is - 7, which means for every 1 unit increase in $x$, $y$ decreases by 7 units. Starting from the point $(0,9)$, if we move 1 unit to the right (increase $x$ by 1), we move 7 units down (decrease $y$ by 7) to the point $(1,2)$. We can then draw a straight line passing through these two points.