Writing & Transforming Linear Functions — Challenge
Once a linear model is written, every question about it — rate, starting value, prediction, shift — is a short calculation.
What these problems cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Arithmetic sequence — A list that grows by adding the same amount each time.
- Function — A rule pairing each input with exactly one output.
- Slope — How much y changes for each 1 that x increases.
- Linear function — One whose graph is a straight line: y = mx + b.
- Rate of change — The slope, in the units of the situation.
- Average rate of change — f(b) - f(a)b - a over an interval.
- Transformation — A change to the function that moves or stretches its graph.
- Sequence — An ordered list of numbers. The nth one is an.
- Explicit formula — A rule giving an directly from n.
- Common difference — The constant added each time, in an arithmetic sequence.
Problem 1
What is the equation of the line through (1,3) and (3,7)?
- y=2x-1
- y=x+2
- y=2x+1
- y=2x+3
- y=12x+1
Worked answer
y=2x+1
The slope is 7-33-1 = 42 = 2.
Substitute (1,3) into y = 2x+b: 3 = 2+b, so b = 1.
The line is y = 2x+1. Check the other point: 2(3)+1 = 7.
Problem 2
A taxi charges $3 to start plus $2 for each mile. Which equation gives the fare y for x miles?
- y=3x+2
- y=2x+3
- y=2x
- y=5x
- y=3x
Worked answer
y=2x+3
The fixed charge of 3 dollars applies whatever the distance, so it is the constant term.
The 2 dollars applies once per mile, so it multiplies x and is the slope.
The equation is y = 2x+3. Swapping the two numbers would charge 3 dollars a mile.
Problem 3
What is the equation of the line through (2,-1) with slope -3?
- y=-3x-5
- y=-3x+7
- y=-3x+5
- y=3x-7
- y=-3x-1
Worked answer
y=-3x+5
Start from y = -3x+b and substitute the point: -1 = -3(2)+b = -6+b.
So b = 5 and the line is y = -3x+5.
Check: at x=2, y = -6+5 = -1 as required.
Problem 4
What is the equation of the line parallel to y=5x-2 that passes through (0,4)?
- y=5x-4
- y=-5x+4
- y=4x+5
- y=5x+4
- y=15x+4
Worked answer
y=5x+4
Parallel lines have equal slopes, so the slope is 5.
The point (0,4) is on the y-axis, so the intercept is 4.
The line is y = 5x+4.
Problem 5
What is the equation of the line perpendicular to y=2x+1 that passes through the origin?
- y=2x
- y=-12x
- y=12x
- y=-2x
- y=-12x+1
Worked answer
y=-12x
Perpendicular slopes are negative reciprocals: flip the fraction and change the sign.
The slope 2 becomes -12.
Through the origin the intercept is 0, so the line is y = -12x.
Problem 6
A tank holds 50 litres and loses 4 litres a minute. Which equation gives the amount y left after x minutes?
- y=4x+50
- y=50x-4
- y=50-4x
- y=-50+4x
- y=4x-50
Worked answer
y=50-4x
At x=0 the tank is full, so the constant term is 50.
The amount falls by 4 each minute, so the rate is -4 and the equation is y = 50-4x.
A positive slope would describe a tank filling up rather than draining.
Problem 7
What is the equation of the line through (0,0) and (5,10)?
- y=12x
- y=2x
- y=5x
- y=10x
- y=x+5
Worked answer
y=2x
The slope is 10-05-0 = 2.
The line passes through the origin, so the intercept is 0.
The equation is y = 2x.
Problem 8
What is the equation of the line with slope -7 and y-intercept 0?
- y=7x
- y=-7x
- y=-7x+1
- y=-17x
- y=x-7
Worked answer
y=-7x
In y = mx+b, here m = -7 and b = 0.
Adding zero changes nothing, so the equation is simply y = -7x.
A zero intercept means the line passes through the origin.