Question

1. given $f(3) = -2$ and $f(5) = 5$, what is the $y$-intercept of $f(x)$?

Explanation:

Step1: Assume linear function

Assume \( f(x) = mx + b \), where \( m \) is slope and \( b \) is y - intercept. We know two points: when \( x = 3 \), \( f(3)=-2 \), so the point is \( (3, - 2) \); when \( x = 5 \), \( f(5) = 5 \), so the point is \( (5,5) \).

Step2: Calculate the slope \( m \)

The formula for slope between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting \( (x_1,y_1)=(3,-2) \) and \( (x_2,y_2)=(5,5) \), we get \( m=\frac{5-(-2)}{5 - 3}=\frac{5 + 2}{2}=\frac{7}{2} \).

Step3: Find the y - intercept \( b \)

Substitute \( m=\frac{7}{2} \) and the point \( (3,-2) \) into \( f(x)=mx + b \). So \( -2=\frac{7}{2}\times3 + b \). Calculate \( \frac{7}{2}\times3=\frac{21}{2} \), then the equation becomes \( -2=\frac{21}{2}+b \). Solve for \( b \): \( b=-2-\frac{21}{2}=\frac{-4 - 21}{2}=-\frac{25}{2}=-12.5 \)

Answer:

The y - intercept of \( f(x) \) is \( -\frac{25}{2} \) (or - 12.5)