How to explain it
At this standard, students extract the rate of change (slope) and initial value (y-intercept) from functions shown as tables, equations, graphs, and verbal descriptions, then compare those properties across two functions given in different representations.
The anchor students hold onto: Table: Δy÷Δx = slope; output at x=0 = y-int. Equation y=mx+b: m=slope, b=y-int. Graph: rise÷run = slope; y-axis crossing = y-int. Verbal: rate = slope, starting value = y-int.
Students use comparison skills in 8.F.B.4 to construct linear functions from two points or a table, and in 8.F.B.5 to sketch graphs that match qualitative descriptions.
Worked examples
Example 1
Table vs Equation
Table(m=3) vs y=2x+5. Rate?
Step 1Table: x=0,1,2,3→y=1,4,7,10. Slope=(4−1)÷1=3.
Step 2Equation y=2x+5: slope m=2.
Step 3Compare: 3>2. Table has greater rate of change.
AnswerTable slope 3 > equation slope 2 — table wins.
Example 2
Equation vs Graph
Eq y=4x−1 vs graph(y-int=3).
Step 1Equation y=4x−1: slope=4, y-int=−1.
Step 2Graph: slope=2, y-int=3 (crosses y-axis at 3).
Step 3Compare y-intercepts: 3>−1. Graph wins.
AnswerGraph y-int 3 > equation y-int −1 — graph wins.
Example 3
Verbal vs Equation
Verbal $50/hr vs y=40x+30.
Step 1Verbal: rate $50/hr→slope=50; fee $20→y-int=20.
Step 2Equation y=40x+30: slope=40, y-int=30.
Step 3Compare: 50>40. Verbal has greater rate.
AnswerVerbal slope 50 > equation slope 40 — verbal wins.
Common mistakes
What students write
Reading slope from a table as the y-value when x=1 rather than computing Δy÷Δx.
The fix
Slope from a table = Δy÷Δx between any two rows. The value at x=1 is NOT the slope.
Try this
Marcus compared f(x)=2x+6 and g(x), shown by a graph through (0,4) with slope 5. Marcus said: "f(x) starts at 2 because 2 is its coefficient." Identify Marcus’s error and state the correct initial value for each function.
What students write
Assuming the function with the larger coefficient has the greater slope without checking the form.
The fix
Extract slope from each form first — m in y=mx+b IS slope, but slope from a table requires Δy÷Δx.
Teacher tip
Head off the two predictable errors before they happen. First: Slope from a table = Δy÷Δx between any two rows. The value at x=1 is NOT the slope. Second: Extract slope from each form first — m in y=mx+b IS slope, but slope from a table requires Δy÷Δx.