Lecture: Linear Functions & Their Graphs
Lecture script · about 15–20 minutes spoken · Week 4
Opening
Think about any price that has a starting fee plus a steady charge. A small-business owner prices an order with a base fee plus a per-item charge. A nursing student reads a drip that adds the same amount each hour. A biology student tracks a seedling that grows the same amount each week. Each of those is a straight line, and once you can read a line's starting value and its rate, you can predict the next number and check it yourself.
Here's where we are in this course. In Week 1 you learned the rules for numbers and signs, so a negative times a negative no longer surprises you. In Week 2 you solved linear equations by doing the same thing to both sides. In Week 3 you met functions, and you priced bowls at a bowl shop with the rule g of x equals 3x plus 4. In this lecture we graph that rule and read its two numbers. Next week, in Week 5, we put two lines on one graph and ask where they meet.
I'll take eight ideas in order: equal steps, slope, the sign of a slope, two forms of a line's equation, intercepts, and how two lines can be parallel or perpendicular. Most examples come from the bowl shop, with a snack budget, a road trip, and a phone plan along the way.
1. Spot a steady rate: equal steps make a line
Let's start at the bowl shop. A plain bowl costs 4 dollars, and every topping adds 3 dollars. So zero toppings cost 4 dollars, one costs 7, two cost 10, three cost 13, four cost 16, and five cost 19. Every step is the same 3 dollars.
That rhythm has a name. A first difference is the amount one output in a table increases or decreases from the output before it when the inputs go up in steps of 1. Subtract each price from the next one. Seven minus four is three. Ten minus seven is three. All the way up, every first difference is 3.
Equal steps are what make a graph a straight line. A linear function is a function whose graph is a straight line, written f of x equals m x plus b; its output changes by the same amount, m, every time the input increases by 1. Picture the six prices as dots, toppings across the bottom and dollars up the side. A ruler laid across them touches all six.
Here's a quick test for any table: subtract neighbors. A list like 4, 8, 16, 32 goes up, but its steps are 4, then 8, then 16. Changing steps mean a curve, not a line.
Pause and try this. Three toppings cost 13 dollars at the bowl shop. Without the formula, what do four toppings cost?
Take a breath and try it before you read on.
Here's the answer. Four toppings cost 16 dollars, because every added topping adds the same first difference of 3 dollars to the 13.
2. Measure steepness: slope as rise over run
Now let's put one number on how steep a line is. Think about a road trip at a steady 60 miles per hour. Every hour, your distance climbs by the same 60 miles. That speed is the steepness of your distance line.
The slope is the steepness of a line, written m: the change in y divided by the change in x between any two points on the line. People also say rise over run: the vertical change between two points divided by the horizontal change between them, with the vertical change on top. The rise is the up-and-down change. The run is the left-to-right change.
Let's find the slope of the bowl-shop line from two points. Take zero toppings at 4 dollars and two toppings at 10 dollars. The rise is 10 minus 4, which is 6. The run is 2 minus 0, which is 2. Six divided by two is three. The slope is 3.
Slope is also a rate of change: how much an output changes for each one-unit increase in the input. For a line, the rate of change equals the slope, and it carries units. Here the units are dollars per topping. So the slope of 3 says each topping you add raises your price by 3 dollars.
The most common slip is turning the fraction upside down. Run on top gives 2 over 6, which is one-third, and that's wrong. Rise over run: the y's go on top.
Pause and try this. Use two other points on the bowl-shop line: one topping at 7 dollars and five toppings at 19 dollars. What slope do you get?
Work it out, then keep reading.
Here's the answer. The rise is 19 minus 7, which is 12, and the run is 5 minus 1, which is 4, so the slope is 12 divided by 4, which is 3, the same as before.
3. Read the sign: rising, falling, horizontal, and vertical lines
Imagine walking along a line from left to right. If you climb, the slope is positive. If you walk downhill, the slope is negative. The bowl-shop line climbs, and its slope, 3, is positive.
There are two special lines you'll meet all the time. Think of an unlimited phone plan that costs the same 40 dollars a month, whatever you use. Its graph is perfectly flat. A horizontal line is a flat line on which every point has the same y-value; its equation has the form y equals k for a number k, and its slope is 0 because its rise is 0. Take the points zero comma four and five comma four. The rise is 4 minus 4, which is 0. Zero divided by five is zero. That line is y equals 4.
Now stand the line straight up. A vertical line is an up-and-down line on which every point has the same x-value; its equation has the form x equals k, and its slope is undefined because its run is 0. Take the points six comma zero and six comma four. The run is 6 minus 6, which is 0. Now you'd have to divide 4 by 0, and no number times zero gives four. So the line x equals 6 has an undefined slope, which has no numerical value because finding it would require dividing the rise by a run of 0.
Here's a way to keep them apart. Zero on top is fine; it gives a slope of zero. Zero on the bottom breaks the division.
Pause and try this. One line is y equals 4, and the other is x equals 6. Which one has a slope of zero?
Decide before you read on.
Here's the answer. The line y equals 4 has slope zero, because it's horizontal and its rise is zero; the line x equals 6 is vertical, so its slope is undefined.
4. Read a line's equation: slope-intercept form
Back to the bowl shop. The price starts at 4 dollars for a plain bowl and climbs 3 dollars per topping. Those two facts are the whole story, and one way of writing a line puts both of them in plain view.
Slope-intercept form is the equation of a line written y equals m x plus b, so that the slope m and the y-intercept b can be read off directly. The y-intercept is the point where a graph crosses the vertical y-axis, found by setting x equal to 0; for a line y equals m x plus b, it's the point zero comma b.
So read y equals 3x plus 4. The number glued to x is 3, so the slope is 3. The number standing alone is 4, so the line crosses the y-axis at zero comma four. Then check a point. At two toppings, 3 times 2 is 6, and 6 plus 4 is 10. The point two comma ten is on the line, and the price for two toppings is 10 dollars.
Why is b the crossing point? When x is 0, the m x part is 0, and all that's left is b.
Here's the swap students make most often. In y equals negative 2x plus 5, it's tempting to call 5 the slope because it's the friendlier number. But the slope is always the number glued to x. Find x first, then read its coefficient.
Pause and try this. In y equals negative 2x plus 5, what is the slope, and where does the line cross the y-axis?
Say both answers to yourself first.
Here's the answer. The slope is negative 2, because negative 2 multiplies x, and the line crosses the y-axis at zero comma five, because 5 is the number standing alone.
5. Build a line from two points: point-slope form
Sometimes you only know two points on a line. You can still build the equation in two moves: find the slope, then pin the line to one point.
Point-slope form is the equation of a line written y minus y-sub-one equals m times the quantity x minus x-sub-one, built from one known point on the line and the line's slope m. The little one is a label that says "this point". It is not an exponent.
Let's build the line through negative one comma four and three comma negative four. First, the slope. On top, negative 4 minus 4 is negative 8. On the bottom, 3 minus negative 1 is 3 plus 1, which is 4. Negative eight divided by four is negative two. So m is negative 2.
Second, pin it to the point negative one comma four. That gives y minus 4 equals negative 2 times the quantity x minus negative 1, which is x plus 1. Distribute the negative 2. Negative 2 times x is negative 2x, and negative 2 times 1 is negative 2. So y minus 4 equals negative 2x minus 2. Add 4 to both sides, and y equals negative 2x plus 2.
Now check both points, because an equation you haven't checked isn't finished. At x equals 3, negative 2 times 3 is negative 6, plus 2 is negative 4. At x equals negative 1, negative 2 times negative 1 is 2, plus 2 is 4. Both points land on the line.
Pause and try this. Suppose you had pinned the line to the other point, three comma negative four, instead. Would you get a different equation?
Think about it for a moment before you go on.
Here's the answer. You'd get the same line. The point form becomes y plus 4 equals negative 2x plus 6, and subtracting 4 gives y equals negative 2x plus 2 again.
6. Find the intercepts: standard form and graphing from two points
Here's a budget story. A study group has 12 dollars. Snacks cost 2 dollars each, and drinks cost 3 dollars each. If x counts snacks and y counts drinks, spending the whole budget means 2x plus 3y equals 12.
That's a line written a third way. Standard form is the equation of a line written A x plus B y equals C, with the x-term and the y-term on the left side and a constant on the right, where A and B are not both 0. It's not built for reading the slope. It is built for finding intercepts. The x-intercept is the point where a graph crosses the horizontal x-axis, found by setting y equal to 0 and solving for x.
Two points are enough to draw any line, and the intercepts are the easiest two. Set x to 0, and 3y equals 12, so y is 4. The y-intercept is zero comma four. Set y to 0, and 2x equals 12, so x is 6. The x-intercept is six comma zero. Plot those two points and lay a ruler through them.
As a story, the group can buy 4 drinks and no snacks, or 6 snacks and no drinks.
If you want the slope, solve for y with your Week 2 moves. Subtract 2x from both sides, and 3y equals negative 2x plus 12. Divide every term by 3, and y equals negative two-thirds x plus 4. The slope is negative two-thirds. That fits the story: more snacks leave room for fewer drinks, so the line falls.
Pause and try this. In the snack budget, what does the x-intercept, six comma zero, mean in plain words?
Put it in your own words first.
Here's the answer. It means the group spends the whole 12 dollars on 6 snacks and buys zero drinks, because on the x-axis the number of drinks, y, is zero.
7. Compare two lines: parallel and perpendicular
Picture two rail tracks that run side by side forever, and then the corner of a sheet of paper, where two edges meet in a perfect square corner.
Parallel lines are two different lines in the same plane that never meet; non-vertical parallel lines have equal slopes and different y-intercepts. Start with the bowl-shop line, y equals 3x plus 4, and build the parallel line through negative one comma four. Keep the slope, 3. Pin it to the point: y minus 4 equals 3 times the quantity x plus 1. That gives y minus 4 equals 3x plus 3, so y equals 3x plus 7. Check it: 3 times negative 1 is negative 3, plus 7 is 4. The two lines climb at the same rate, one starting at 4 and one at 7, and they never meet.
Perpendicular lines are two lines that cross at a right angle; when neither line is vertical, their slopes multiply to negative 1. The partner slope has a name. The negative reciprocal is the number found by flipping a nonzero fraction upside down and changing its sign; a number times its negative reciprocal is negative 1. Take two-thirds. Flip it to three-halves, and change the sign to negative three-halves. Two-thirds times negative three-halves is negative six-sixths, which is negative 1.
Two traps live here. If you only change the sign, two-thirds times negative two-thirds is negative four-ninths. If you only flip, two-thirds times three-halves is positive 1. You need both moves.
Pause and try this. A classmate says the line perpendicular to a slope of two-thirds has slope three-halves. Is that right?
Check it with a quick multiplication.
Here's the answer. It isn't, because two-thirds times three-halves is positive 1, not negative 1; the perpendicular slope is negative three-halves.
8. Check perpendicular with arithmetic, not your eyes
Graphing apps draw on screens of every shape. On a stretched window, one step across can be drawn wider than one step up, so angles look different from what they really are.
So in this course, the slopes decide and the picture gives a second opinion. Take slopes of two-thirds and negative four-thirds. On a squashed screen, those lines can look perpendicular. Multiply them: two-thirds times negative four-thirds is negative eight-ninths. That's close to negative 1, but it isn't negative 1. Someone laying out a floor plan with those lines would get a slightly crooked corner.
Here's the honest caveat. A picture depends on the window and the screen, but the product of two slopes depends only on the slopes.
Pause and try this. Two lines have slopes of two-thirds and negative four-thirds, and on your screen they look perpendicular. Would you call them perpendicular?
Decide, and give your reason, before you read on.
Here's the answer. No, because their product is negative eight-ninths, and perpendicular slopes must multiply to exactly negative 1.
Closing
Let's pull it together, in the order we took it. Equal steps in a table, like the bowl shop's steps of 3, make a straight line. The slope is rise over run, with the y's on top, and it's a rate of change with units. A positive slope climbs, a negative one falls, a horizontal line has slope zero, and a vertical line has no slope at all. In y equals m x plus b, the slope is glued to x and the y-intercept stands alone. Point-slope form builds a line from one point and its slope, and you check both points afterward. Intercepts come from setting the other variable to zero. Parallel lines share a slope, and perpendicular slopes multiply to negative 1. When a picture and the arithmetic disagree, trust the arithmetic.
Next in this module, you'll find the slide deck, then the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. In Week 5, Systems of Linear Equations & Inequalities, two of these lines share one graph, and you'll find the one point where they meet.
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