Question

if $f(x)$ is a linear function and given $f(6) = 1$ and $f(9) = 4$, determine the linear function.
a.) what is the slope? \boxed{} (be sure to leave your answer in reduced fraction form.)
b.) what is the $y$-intercept? \boxed{} (be sure to leave your answer in reduced fraction form.)
c.) what is $f(x)$?
$f(x) = \boxed{}$

Explanation:

Part a: Find the slope

Step1: Identify two points

Since \( f(x) \) is linear, \( f(6) = 1 \) means the point \( (6, 1) \) and \( f(9) = 4 \) means the point \( (9, 4) \).

Step2: Use slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting \( x_1 = 6, y_1 = 1, x_2 = 9, y_2 = 4 \), we get \( m=\frac{4 - 1}{9 - 6} \).

Step3: Simplify the fraction

\( \frac{4 - 1}{9 - 6}=\frac{3}{3}=1 \).

Step1: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We know \( m = 1 \), and we can use the point \( (6, 1) \). So the equation becomes \( y - 1=1\times(x - 6) \).

Step2: Simplify to slope - intercept form

Simplify \( y - 1=x - 6 \) by adding 1 to both sides: \( y=x - 6 + 1=x - 5 \). The slope - intercept form is \( y=mx + b \), where \( b \) is the y - intercept. So \( b=- 5=\frac{-5}{1} \).

Step1: Recall linear function form

A linear function is of the form \( f(x)=mx + b \).

Step2: Substitute \( m \) and \( b \)

We found \( m = 1 \) and \( b=-5 \), so \( f(x)=1\times x-5=x - 5 \).

Answer:

\( 1 \)

Part b: Find the y - intercept