8.F.B.4
8th Grade
Functions
Constructing Linear Functions
Construct a function to model a linear relationship between two quantities, determining the rate of change and initial value.
How to explain it
The anchor students hold onto: Two pts: m=(y2-y1)/(x2-x1); sub pt for b. Table: slope=dy/dx; b=y at x=0. Graph: rise/run=m; y-axis=b. Verbal: rate=m, start=b.
Students apply construction skills in 8.F.B.5 to sketch and interpret qualitative graphs, and in 8.EE.B.5-6 to connect slope to proportional relationships and similar triangles.
Worked examples
Example 1
— Two Points
Two points: (1,5) and (3,11).
Step 1Slope: m = (11-5) / (3-1) = 6/2 = 3.
Step 2Use (1,5): 5 = 3(1) + b → b = 2.
Step 3Function: y = 3x + 2.
Answery = 3x + 2 (m = 3, b = 2)
Example 2
— Table
Table: x=0,1,2,3 / y=4,7,10,13.
Step 1Slope: (7-4) / (1-0) = 3.
Step 2Y-int: y = 4 when x = 0 → b = 4.
Step 3Function: y = 3x + 4.
Answery = 3x + 4 (m = 3, b = 4)
Example 3
— Verbal
Plumber: $25/hr, $40 flat fee.
Step 1Rate = $25/hr → slope m = 25.
Step 2Starting fee = $40 → b = 40.
Step 3Function: y = 25x + 40.
Answery = 25x + 40 (m = 25, b = 40)
Common mistakes
What students write
Using y1/x1 as slope — dividing a single y-value by its x-value is NOT the slope formula.
The fix
Slope is a CHANGE ratio: m = (y2-y1)/(x2-x1). Always subtract coordinates.
Try this
A student found the function through (1, 7) and (4, 13) and wrote: m = 7/1 = 7; equation y = 7x + 13. Identify BOTH errors and write the correct equation.
What students write
Reading b as any y-value without checking whether x = 0 at that point.
The fix
Only read b directly when x = 0 is confirmed; otherwise substitute a point and solve.
Teacher tip
Head off the two predictable errors before they happen. First: Slope is a CHANGE ratio: m = (y2-y1)/(x2-x1). Always subtract coordinates. Second: Only read b directly when x = 0 is confirmed; otherwise substitute a point and solve.