Question

1. write a linear function f with the given values:

$f(0) = 2,$ $f(3) = -1$

1. identify the vertex of the following absolute value equation:

$f(x) = |x + 4| - 2$

1. write an equation in slope intercept form of a line parallel to the line $y = \frac{3}{5}x - 8$ and passes though the point $(0, -3)$.
2. write an equation in point slope form of a line perpendicular to the line $y = -2x - 7$ and passes though the point $(-3, 10).$

Explanation:

Problem 38:

Step1: Find the slope \( m \)

The two points are \((0, 2)\) and \((3, -1)\). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substituting the values: \( m=\frac{-1 - 2}{3 - 0}=\frac{-3}{3}=-1 \)

Step2: Find the y - intercept \( b \)

The linear function is in the form \( f(x)=mx + b \). We know that when \( x = 0 \), \( f(0)=2 \). Substituting \( x = 0 \), \( m=-1 \) into \( f(x)=mx + b \):
\( 2=-1\times0 + b \), so \( b = 2 \)

Step3: Write the linear function

Substitute \( m=-1 \) and \( b = 2 \) into \( f(x)=mx + b \), we get \( f(x)=-x + 2 \)

The general form of an absolute - value function is \( f(x)=|x - h|+k \), where \((h,k)\) is the vertex of the graph of the function.
For the function \( f(x)=|x + 4|-2 \), we can rewrite \( x + 4\) as \( x-(-4) \). Comparing with \( f(x)=|x - h|+k \), we have \( h=-4 \) and \( k=-2 \)

Step1: Determine the slope of the parallel line

If two lines are parallel, they have the same slope. The given line is \( y=\frac{3}{5}x-8 \), which is in slope - intercept form \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). So the slope of the given line is \( m=\frac{3}{5} \), and the slope of the line parallel to it is also \( \frac{3}{5} \)

Step2: Determine the y - intercept

The line passes through the point \((0,-3)\). In the slope - intercept form \( y=mx + b \), when \( x = 0 \), \( y=b \). So \( b=-3 \)

Step3: Write the equation of the line

Substitute \( m=\frac{3}{5} \) and \( b=-3 \) into \( y = mx + b \), we get \( y=\frac{3}{5}x-3 \)

Answer:

\( f(x)=-x + 2 \)

Problem 39: