Use this answer key to score the linear functions test out of 20 points. Each solution shows the work and a scoring guide. Give partial credit for correct steps even when the final answer has an arithmetic slip.
Question 1: Rate and initial value (2 points)
(a) The rate of change is \(-4\) gallons per minute (the tank loses 4 gallons each minute).
(b) The initial value is \(30\) gallons.
Scoring: 1 point for -4 with units, 1 point for 30 gallons.
Question 2: Equation from a table (2 points)
\(y\) increases by 4 for each step of 1, so \(m=4\). Then \(9=4(1)+b\) gives \(b=5\).
Equation: \(y=4x+5\).
Scoring: 1 point for the rate 4, 1 point for b = 5 and the equation.
Question 3: Equation from two points (3 points)
\(m=\frac{11-(-1)}{4-(-2)}=\frac{12}{6}=2\).
\(-1=2(-2)+b\) gives \(-1=-4+b\), so \(b=3\).
Equation: \(y=2x+3\). Check: \(2(4)+3=11\).
Scoring: 1 point for m = 2, 1 point for b = 3, 1 point for the equation written correctly.
Question 4: Equation from a graph (3 points)
The line crosses the \(y\)-axis at 5, so \(b=5\). From \((0,5)\) to \((2,1)\) the run is 2 and the rise is \(-4\), so \(m=\frac{-4}{2}=-2\).
Equation: \(y=-2x+5\).
Scoring: 1 point for b = 5, 1 point for the rate -2, 1 point for the equation.
Question 5: Linear or nonlinear? (2 points)
(a) The changes in \(y\) are 3, 3, 3: constant, so P is linear.
(b) The changes are 1, 2, 3: not constant, so Q is nonlinear.
Scoring: for each table, 1 point for the correct decision supported by the changes.
Question 6: Club savings (2 points)
(a) \(y=12x+85\).
(b) \(12x+85=205\), so \(12x=120\) and \(x=10\). 10 weeks.
Scoring: 1 point for the function, 1 point for 10 weeks.
Question 7: Describe a graph (2 points)
From 0 to 4 seconds the height is increasing (the elevator goes up 12 feet, \(3\) feet per second). From 4 to 8 seconds it is constant at 12 feet (stopped). From 8 to 12 seconds it is decreasing back to the ground (\(-3\) feet per second).
The whole graph is not linear because it is not one straight line.
Scoring: 1 point for the three parts described correctly, 1 point for saying the graph is nonlinear with a reason.
Question 8: Sketch a graph (2 points)
The graph rises in a straight line, then is horizontal, then falls in a steeper straight line to the ground.
Scoring: 1 point for rising then level, 1 point for a descent that is steeper than the rise and ends at the ground.
Question 9: Filling a pool (2 points)
(a) \(m=\frac{230-150}{5-3}=\frac{80}{2}=40\) gallons per minute.
(b) \(150=40(3)+b\) gives \(b=30\). The function is \(y=40x+30\), and the initial value is 30 gallons.
Scoring: 1 point for 40 gallons per minute, 1 point for the equation with initial value 30.
20 points
See every chapter on the Grade 8 math page.