Algebra 1
Unit 4 · Lesson 4.3

Multiple representations; linear vs. nonlinear

A function rarely shows up in the form you'd have picked. Sometimes it's a sentence, sometimes a table, sometimes an equation, sometimes a picture. The useful skill is seeing that those are four views of one and the same relationship, and being able to slide from any one to any other. This is exactly the fluency the next unit builds on when it starts graphing lines.

Start with a plain sentence: start with $1, and add $2 every step. That's a relationship between a step number and an amount of money, and you can lay it out as a table by just following the instruction:

x 0 1 2 3
f(x)=2x+1 1 3 5 7

At step 0 you have the starting $1. Each step adds $2: 1, then 3, then 5, then 7. The same relationship has an equation too, f(x) = 2x + 1, and if you plot the table's pairs as points, they fall in a straight line. Words, table, equation, graph: one relationship, four faces.

Look closely at the output row. From 1 to 3 is +2; from 3 to 5 is +2; from 5 to 7 is +2. Equal steps in x (here, +1 each time) produce equal steps in the output (+2 each time). That steady, repeating step is what your eye reads as "straight" when the points are plotted, and it has a name: a constant rate of change. A function whose rate of change is constant, equal x-steps always giving equal output-steps, is a linear function, and its graph is a straight line.

Not every relationship is so steady. Watch what happens with y = x²:

x 0 1 2 3
y=x^2 0 1 4 9

The differences are 1 − 0 = 1, then 4 − 1 = 3, then 9 − 4 = 5: the steps go 1, 3, 5, and they keep growing. The rate of change isn't constant, so this is a nonlinear function, and its graph curves instead of running straight.

Here's the line and the curve side by side 4.3.f1, so you can see "straight versus curved" at a glance. The curve has a name, a parabola, but for now the point is just that changing steps mean a bend, not a line.

x y (0,1) y = 2x + 1
Figure 4.3.f1

That comparison is the heart of it, so here are the two graphs together once more, with their difference rows underneath 4.3.f2: a constant difference draws a straight line, and a growing difference draws a curve.

x y vertex (0,0) (2,4)
Figure 4.3.f2

One thing the constant-difference shortcut relies on: it only works when the x-values step evenly, so check the inputs are evenly spaced first, then read the output differences.

The four faces are meant to be reversible, not just read left to right. You can go from equation to table by plugging in x-values, and you can go from table back to equation by reading the steady step. For a linear table, that backward move is short: the constant step is the number that sits in front of x, and the output at x = 0 is the constant added on. Once you've recovered the equation, check it against one row of the table to be sure.

New words

  • 4.3.d1 Linear function: constant rate of change; its graph is a straight line; its equation has the form y=mx+b (equivalently f(x)=mx+b), where (looking ahead to Unit 5) m is that constant step (how much the output changes per +1 in x) and b is the starting output at x=0. For now you don't need to manipulate m and b; just recognize the straight-line shape and the constant step.
  • 4.3.d2 Constant rate of change: equal steps in x always produce equal steps in the output. (This is the unit rate from Unit 3, and it becomes slope in Unit 5.)
  • 4.3.d3 Nonlinear function: rate of change is not constant; the graph curves (e.g. y=x²).

Read each example a line at a time. The move to study here is reading the constant step off a table.

Worked example

4.3.w1 Example 1: build a table from f(x)=2x+1; note the constant step.

x 0 1 2 3
f(x)=2x+1 1 3 5 7

Output differences: 3-1=2, 5-3=2, 7-5=2, so it's constant +2 for each +1 in x. Linear. That constant +2 is the rate of change (the future slope).

4.3.w2 Example 2: contrast with y=x².

x 0 1 2 3
y=x^2 0 1 4 9

Differences: 1-0=1, 4-1=3, 9-4=5, giving 1,3,5, not constant. Nonlinear (this curve is a parabola).

4.3.w3 Example 3: linear-or-not from a table (no equation given).

x 1 2 3 4
y 1 4 7 10

Equal x-steps of +1; output steps 3,3,3, so it's constant. Linear. (Rule: y=3x-2.)

4.3.w4 Example 4: same relationship, four faces (translate in both directions).

  • Words: "start at $1, add $2 each step."
  • Equation: f(x)=2x+1.
  • Table: the table from Example 1.
  • Graph: a straight line through (0,1) rising 2 for every 1 right.

All four describe one function, and the skill runs both ways: given the equation you can build the table (plug in x=0,1,2,3 to get 1,3,5,7), and given the table you can read off the equation (the constant step +2 is the number in front of x, and the value at x=0 is the +1, so f(x)=2x+1). Practice going each direction, not just left-to-right.

4.3.w5 Example 5: both directions explicitly (table ↔ equation).

  • Equation → table. From g(x)=5x−2, build the table at x=0,1,2,3: g(0)=−2, g(1)=3, g(2)=8, g(3)=13.
x 0 1 2 3
g(x)=5x−2 −2 3 8 13
  • Table → equation. Now go the other way. Given this table
x 0 1 2 3
y 2 5 8 11

read the constant step (5−2=3, 8−5=3, 11−8=3 → +3 per step, so 3 sits in front of x) and the start at x=0 (that's 2), giving y=3x+2. Check it against the table: 3(2)+2=8. Same relationship, recovered from numbers alone.

After all that translating, here's a clean one to settle the idea. Outputs go 2, 2, 2, 2 for x = 0, 1, 2, 3. The steps are 0, 0, 0, which is constant, so it's linear, with a rate of change of 0. A perfectly flat, steady relationship is still linear; the steady step just happens to be nothing.

A couple of slips to know about, now that you've done the real thing a few times. The first sounds harmless: deciding a relationship is linear because the outputs climb by the same amount, without first checking that the inputs step evenly. Equal output-jumps only mean a straight line when they're measured over equal input-steps, so look at the x-row first.

The second is a worry that the curve from Example 2 somehow "failed." It didn't. A curve is still a perfectly good function. It passes the vertical line test from Lesson 4.1; it simply isn't a straight one. Nonlinear isn't broken; it's just not a line.

Check yourself

  1. 4.3.c1 Here's a table with x = 0,1,2,3 and y = 2,2,2,2. Linear or not, and what's its rate of change? (Linear: the output steps are 0, 0, 0, which is constant, and the rate of change is 0.)
  2. 4.3.c2 Turn the words "start at 10 and lose 1 each step" into an equation and a four-row table. (Equation: y = 10 − x. Table at x=0,1,2,3: outputs 10, 9, 8, 7.)
  3. 4.3.c3 A table's outputs go 2, 6, 12, 20 for x=1,2,3,4. Linear or nonlinear, and how can you tell without graphing? (The x-steps are even, +1 each. The output steps are 4, 6, 8, not constant, so it's nonlinear; you can tell straight from the changing differences.)

These mix building tables, classifying, and translating both ways, so expect to switch gears between them. Each answer is at the end of the lesson. If one stalls you, find the worked example it's based on.

Practice

A. Fill the table from the rule.

4.3.1 f(x)=3x: find outputs at x=0,1,2,3,4.
Reveal answerHide to problem 10,3,6,9,12 — constant +3, linear.
4.3.2 k(x)=10-x: find outputs at x=0,1,2,3.
Reveal answerHide to problem 210,9,8,7 — constant -1, linear.
4.3.3 f(x)=2x+1: find outputs at x=-2,-1,0,1.
Reveal answerHide to problem 3-3,-1,1,3 — constant +2, linear.

B. Linear or nonlinear? (equal x-steps; check the output differences)

4.3.4 x:0,1,2,3 → y:1,4,7,10
Reveal answerHide to problem 4Linear — output steps 3,3,3.
4.3.5 x:0,1,2,3 → y:0,1,4,9
Reveal answerHide to problem 5Nonlinear — steps 1,3,5 (this is y=x²).
4.3.6 x:1,2,3,4 → y:5,5,5,5
Reveal answerHide to problem 6Linear — steps 0,0,0 (a constant/horizontal function; rate of change 0).
4.3.7 x:0,1,2,3 → y:1,2,4,8
Reveal answerHide to problem 7Nonlinear — steps 1,2,4 (doubling).

C. Match the representation.

4.3.8 Which equation matches the words "start at $1 and add $2 each step": (a) f(x)=x+2, (b) f(x)=2x+1, (c) f(x)=2x?
Reveal answerHide to problem 8(b) f(x)=2x+1 — start 1 at x=0, add 2 per step.

D. Translate both directions (table ↔ equation).

4.3.9 Table → equation. The table x:0,1,2,3 → y:2,5,8,11 is linear. Write its equation.
Reveal answerHide to problem 9y=3x+2 — constant step +3 (so 3 in front of x), value 2 at x=0. Check: 3(3)+2=11.
4.3.10 Equation → table. From g(x)=5x-2, fill the outputs at x=0,1,2,3.
Reveal answerHide to problem 10-2, 3, 8, 13 — at x=0,1,2,3 (constant step +5, linear).

You can now decide whether a pairing is a function and run the vertical line test on a graph; read and use f(x) without mistaking it for multiplication, evaluate at numbers and at negatives without losing a sign, and give a domain and range; and move one relationship between words, a table, an equation, and a graph, telling a straight line from a curve by whether its rate of change holds steady.

A few threads worth carrying forward. Every time you evaluated a function, you were also practicing order of operations and signed arithmetic from Units 1 and 2, which is why those keep mattering. The constant rate of change from this lesson is the same unit rate from Unit 3, and in Unit 5 it gets one more name, slope. A good warm-up before that unit: build a table from f(x) = mx + b and read off its constant step.

From here on, when a line or a curve shows up, it helps to hear it as a function. The line is a function, and f(x) = 2x + 1 just gives it a name.