Question

lesson 16 • session 3
writing an equation for a linear function from
two points
write an equation for each linear function. show your work.
1 the function that passes through the points (24, 28) and (8, 8)
2 the function represented by the table:

x	y
6	15
2	13

3 the function represented by the table:

input	10	5	-5
output	12	9	3

4 the function that passes through the points (-2, 6) and (2, 7)
5 image of a line on a coordinate grid
6 image of a line on a coordinate grid
grade 8 • lesson 16

Explanation:

Problem 1: Function through (24, 28) and (8, 8)

Step1: 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 \((x_1, y_1)=(24, 28)\) and \((x_2, y_2)=(8, 8)\):
\(m=\frac{8 - 28}{8 - 24}=\frac{-20}{-16}=\frac{5}{4}\)

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((8, 8)\) and \(m = \frac{5}{4}\):
\(y - 8=\frac{5}{4}(x - 8)\)

Step3: Simplify to slope - intercept form (\(y=mx + b\))

\(y-8=\frac{5}{4}x-10\)
\(y=\frac{5}{4}x - 10 + 8\)
\(y=\frac{5}{4}x-2\)

Step1: Calculate the slope (\(m\))

Using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \((x_1, y_1)=(6, 15)\) and \((x_2, y_2)=(2, 13)\):
\(m=\frac{13 - 15}{2 - 6}=\frac{-2}{-4}=\frac{1}{2}\)

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((6, 15)\) and \(m=\frac{1}{2}\):
\(y - 15=\frac{1}{2}(x - 6)\)

Step3: Simplify to slope - intercept form

\(y-15=\frac{1}{2}x - 3\)
\(y=\frac{1}{2}x-3 + 15\)
\(y=\frac{1}{2}x + 12\)

Step1: Calculate the slope (\(m\))

Using the points \((x_1, y_1)=(10, 12)\) and \((x_2, y_2)=(5, 9)\):
\(m=\frac{9 - 12}{5 - 10}=\frac{-3}{-5}=\frac{3}{5}\)
We can verify with another pair \((5, 9)\) and \((-5, 3)\):
\(m=\frac{3 - 9}{-5 - 5}=\frac{-6}{-10}=\frac{3}{5}\)

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((10, 12)\) and \(m=\frac{3}{5}\):
\(y - 12=\frac{3}{5}(x - 10)\)

Step3: Simplify to slope - intercept form

\(y-12=\frac{3}{5}x-6\)
\(y=\frac{3}{5}x-6 + 12\)
\(y=\frac{3}{5}x + 6\)

Answer:

\(y=\frac{5}{4}x - 2\)

Problem 2: Function represented by the table (\(x:6,2\); \(y:15,13\))