use the graph of the line shown.\na. write a point - slope form of the equation for the line shown.\nb…

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope. Given the points $(- 5,4)$ and $(5,1)$, the slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1 - 4}{5+5}=-\frac{3}{10}$. Using the point $(5,1)$, the point - slope form is $y - 1=-\frac{3}{10}(x - 5)$.

Step2: Find y - intercept estimate

By visual inspection of the graph, the y - intercept is about 2.5.

Step3: Use proportional reasoning

The line crosses the y - axis halfway between the given points in the x - direction. The y - values of the points are 4 and 1. The number halfway between 4 and 1 is $\frac{4 + 1}{2}=2.5$.

Step4: Convert to slope - intercept form

Starting with $y - 1=-\frac{3}{10}(x - 5)$, expand the right side: $y-1=-\frac{3}{10}x+\frac{3}{2}$. Then add 1 to both sides: $y=-\frac{3}{10}x+\frac{3}{2}+1=-\frac{3}{10}x+\frac{3 + 2}{2}=-\frac{3}{10}x+\frac{5}{2}=-\frac{3}{10}x + 2.5$. The y - intercept from the slope - intercept form $y=mx + b$ (where $b$ is the y - intercept) is 2.5, which confirms our conjecture.

Answer:

a. $y - 1=-\frac{3}{10}(x - 5)$ b. 2.5 c. The line crosses the y - axis halfway between the given points so the y - intercept is halfway between the y - values of the point on the left, 4, and that of the point on the right, 1. The number halfway between those values is 2.5. d. $y=-\frac{3}{10}x+2.5$