Week 1: Lines – Day 5:
True or False?
1. The graph of a straight line in the x-y plane can have a y-intercept or an x-intercept, but not both.
False. Counterexample: the line y = -x + 1 has y-intercept of (0, 1) and an x-intercept of (1, 0).
2. If F(x) = 2x + 1, then:
F(z) + F(w) = F(z + w)
False. F(z) + F(w) = 2z + 1 + 2w + 1 = 2z + 2w + 2
But F(z + w) = 2(z + w) + 1 = 2z + 2w + 1
2z + 2w + 2 does not equal 2z + 2w + 1, no matter what z and w are.
3. If G(x) = 3x, then:
G(z) + G(w) = G(z + w)
True. G(z) + G(w) = 3z + 3w = 3(z + w) = G(z + w)
4. It is impossible for a line with slope 10 to pass through the point (5, 5).
False. The line y – 5 = 10(x – 5) which corresponds to y – 5 = 10x – 50 or y = 10x – 45 passes through (5, 5).
There are lines of every possible slope passing through every point.
5. If two lines in the x-y plane have two points in common, they have all points in common.
True. All it takes is two points to define a line, and if the two points of Line One and Line Two are the same, then Line One and Line Two are the same line.
Week 1: Lines – Day 4:
1. An alien space baby is born on Year Zero. At birth, the baby is 10 cm tall. The baby grows 1 cm taller per month, or 12 cm per year. If x is the time in years after Year Zero, write an equation that describes the height of the baby as a function of the year.
h(x) = 12x + 10
2. How tall is the alien space baby of problem 1 at the age of 5 years?
h(5) = 12*5 + 10 = 70 cm
3. How tall is the alien space baby of problem 1 at the age of 30 years?
h(30) = 12*30 + 10 = 360 + 10 = 370
The space baby is very tall at 30 – over ten feet.
Bonus question: if the alien space baby looks human, when would people start to suspect that the alien space baby is an alien?
Human growth tends to slow down in childhood, around age 4, and then speed up around age 13, and then slow down to close to zero by age 18. People might start to observe the strangeness of the space baby’s growth pattern as early as age 4, but more likely around age 20, when the baby is still growing.
4. Betty’s sequoia tree grows one ring per year. Let x be the current year (2014). Betty takes a core sample and finds out that her tree has 157 rings in the year 2014. Write an equation that describes the number of rings in the sequoia tree as a function of the calendar year x.
R(x) = (x – 2014) + 157 = x – 1857
Bonus: what year was the sequoia tree born?
R(x) = 0 when x = 1857. The tree was born in 1857.
5. The equation for the number of rings in Betty’s sequoia tree as a function of calendar year, when you graph it, is the equation of a straight line. Does Betty’s sequoia tree grow in a manner similar to the alien space baby? Why or why not?
Some things to think about: how does a typical human grow? What would your own human height as a function of age look like, compared to the alien space baby’s? What are some different ways to measure the growth of a tree? What if, instead of rings, you measured the tree’s gain in diameter each year – what kind of function would that be?
The rings of the tree increase steadily like the height of the space baby. But the growth of the tree does slow down; tree rings get progressively thinner. When the tree is young, it grows by large fractions; for example, a young sapling might grow tenth or more of its height in a year. When the tree is over a hundred years old, it might only grow a hundredth of its height in a year. I’d say the tree does not really grow in a manner similar to the alien space baby, because its growth slows down.
Week 1: Lines – Day 3:
1. Is the point (0, 0) on, above, or below the line y = 3x + 2?
Plugging in x=0 to the equation of the line gives y = 2. So (0, 0) is below the line.
Alternatively you can sketch the graph and see the answer.
2. Is the point (-1, 5) on, above, or below the line x + y = 4?
If x = -1, on the line we have – 1 + y = 4, which means y = 5 is on the line. (-1, 5) is on the line.
3. Do the lines y = 17x – 1 and y = 15x – 1 intersect?
The two lines have different slopes: 17 and 15. They are not parallel and do intersect.
4. Are the points (2, 3), (-5, 10) and (0, 5) collinear? (Collinear means they all lie on the same line.)
There are many ways to answer this question. One is to notice that all three points satisfy the equation x + y = 5, which is the equation of a line. Another way is to calculate the slope of the lines joining the points. In every combination, the slope comes out to -1.
5. Are the points (1, 4), (2, 8), and (3, 13) collinear?
If you calculate the slope from the first two points, you get m = 4. If you calculate the slope from the last two points, you get m = 5. So they can’t be on the same line. Another way to answer this question is to look at the pattern of the first two points: y = 4x. The third point breaks this pattern. Yet another method is to compute the equation of the line joining two of the points, and showing that the third does not fit.
Week 1: Lines – Day 2:
1. What is the equation of the line through the points (0, 0) and (4, 3)?
Oops! Correction: Slope is: (3 – 0)/(4 – 0) = 3/4; y = (3/4)x
2. What is the equation of the line through (0, 5) and (-1, 7)?
y-intercept is (0, 5) and slope m = (7 – 5)/(-1 – 0) = -2;
y = -2x + 5
3. What is the equation of the line through (3, 3) and (6, 3)?
Horizontal line (m = 0): equation is y = 3
4. What is the equation of the line that has a slope of 4 and a y-intercept of (0, 10)?
y = 4x + 10
5. What is the equation of the line through (7, 5) and (7, 10)?
Vertical line (m is undefined): x = 7
Week 1, Day 1:
2. the line 3x – y = 2 is steeper; it corresponds to y = 3x + 2 and has a slope of 3. The other line corresponds to y = (1/3)x + 4 and has a lower slope of 1/3.
3. The slope of the given line is 7/2 or 3.5. Any parallel line has the same slope, 7/2.
4. Any line parallel to y = 3 – 5x has to have a slope of -5. So some example solutions are:
y = -5x ; y = 1 – 5x ; y = -5x + 1003
5. To see if the point (2, 10) is on, above, or below the line y – 4x = 2 plug in x = 2:
y – 4*2 = y – 8
The equation says that y – 4x has to be 2, therefore:
– if y = 10, y – 4x = 2, and the point is on the line: (2, 10) is on the line.
– if y > 10, the point is above the line – for example, (2, 11) is above the line.
– if y < 10, the point is below the line – for example, (2, 9) is below the line.