Question

sections 4-1 to 4-3
period
be sure to show/explain how you arrived at your answer
use the given information to write an equation of the line in
slope-intercept form.

1. slope -4, y-intercept: 9

2.

1. (-2, -6) & (2, -1)

Explanation:

Problem 1:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Step2: Substitute values

Given that the slope $m=-4$ and the $y$-intercept $b = 9$. Substitute these values into the slope - intercept form: $y=-4x + 9$.

Step1: Find the slope

We have two points on the line: $(0,-1)$ (the $y$-intercept) and $(3,-2)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0,-1)$ and $(x_2,y_2)=(3,-2)$. Then $m=\frac{-2-(-1)}{3 - 0}=\frac{-2 + 1}{3}=\frac{-1}{3}$.

Step2: Identify the $y$-intercept

The $y$-intercept $b$ is the value of $y$ when $x = 0$. From the point $(0,-1)$, we know that $b=-1$.

Step3: Write the equation

Using the slope - intercept form $y=mx + b$, substitute $m =-\frac{1}{3}$ and $b=-1$. So the equation is $y=-\frac{1}{3}x-1$. (Note: There was a mistake in the original hand - written answer. The correct $y$-intercept from the point $(0,-1)$ is $-1$, not $1$)

Step1: Calculate the slope

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Given the points $(-2,-6)$ and $(2,-1)$, let $(x_1,y_1)=(-2,-6)$ and $(x_2,y_2)=(2,-1)$. Then $m=\frac{-1-(-6)}{2-(-2)}=\frac{-1 + 6}{2 + 2}=\frac{5}{4}$.

Step2: Find the $y$-intercept

Use the point - slope form $y - y_1=m(x - x_1)$ and then convert to slope - intercept form. Let's use the point $(2,-1)$. Substitute $m=\frac{5}{4}$, $x_1 = 2$ and $y_1=-1$ into $y - y_1=m(x - x_1)$:
$y-(-1)=\frac{5}{4}(x - 2)$
$y + 1=\frac{5}{4}x-\frac{5}{2}$
Subtract $1$ from both sides: $y=\frac{5}{4}x-\frac{5}{2}-1=\frac{5}{4}x-\frac{5 + 2}{2}=\frac{5}{4}x-\frac{7}{2}$

Answer:

$y=-4x + 9$

Problem 2: