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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

First and Second Differences: Recognizing a Quadratic Relation

1.2About 65 minutes
Learning goals
  • I can calculate first and second differences from a table of values.
  • I can tell whether a relation is linear, quadratic or neither from its differences.
  • I can explain why the x-values must go up in equal steps.
  • I can connect the second difference to the value and sign of a in y = ax² + bx + c.

Words to know

first differences
The changes in y between consecutive rows of a table whose x-values go up in equal steps.
second differences
The changes between consecutive first differences.
linear relation
A relation with constant first differences; its graph is a straight line.
quadratic relation
A relation with constant, non-zero second differences; its graph is a parabola.
standard form
The form y = ax² + bx + c of a quadratic relation, where a ≠ 0.
exponential relation
A relation in which each y-value is the previous one multiplied by the same number, such as y = 2ˣ.

A detective's test for tables

Suppose someone hands you a table of values with no graph and no equation. Can you tell what kind of relation it is? In Grade 9 you learned one test: if the first differences are constant, the relation is linear. Today you will learn the next test. If the first differences are not constant but the second differences are, the relation is quadratic.

First and second differences

Make sure the x-values go up by equal steps (for example 1, 2, 3, 4 or 0, 0.5, 1, 1.5). Then:

  • First differences are the changes in y from one row to the next: next y − this y.
  • Second differences are the changes in the first differences: next first difference − this first difference.

For a linear relation the first differences are constant (so the second differences are all 0). For a quadratic relation the first differences change, but by the same amount every time, so the second differences are constant and not zero.

xy = x²1st diff.2nd diff.−39−24−1100112439−5−3−1+1+3+5+2+2+2+2+2
The table for y = x². The first differences (−5, −3, −1, 1, 3, 5) are not constant, but they go up by 2 every time. The second differences are all 2: y = x² is quadratic.
xy0123012345678910up 1up 3up 5over 1 each time
The same idea on the graph of y = x². Moving over 1 each time from the vertex, the graph goes up 1, then 3, then 5. The rises grow by 2 each step, which is why the graph keeps getting steeper and curves.
Worked example

Example 1: Differences for y = 2x² − 3x + 1

Make a table of values for y = 2x² − 3x + 1 for x = 0, 1, 2, 3, 4. Find the first and second differences.

  1. Substitute each x. Use the order of operations: square first, then multiply, then add and subtract.
  2. x = 0: 2(0)² − 3(0) + 1 = 1. x = 1: 2(1) − 3 + 1 = 0. x = 2: 2(4) − 6 + 1 = 3. x = 3: 2(9) − 9 + 1 = 10. x = 4: 2(16) − 12 + 1 = 21.
  3. First differences: 0 − 1 = −1, 3 − 0 = 3, 10 − 3 = 7, 21 − 10 = 11.
  4. Second differences: 3 − (−1) = 4, 7 − 3 = 4, 11 − 7 = 4.

y = 1, 0, 3, 10, 21. The first differences −1, 3, 7, 11 are not constant; the second differences are all 4, so the relation is quadratic.

A pattern worth knowing

In Example 1, a = 2 and the second difference is 4. For y = x², a = 1 and the second difference is 2. When the x-values go up by 1, the second difference of y = ax² + bx + c is always 2a. So the second difference tells you the value of a, and its sign tells you which way the parabola opens: positive means it opens up, negative means it opens down.

Are the x-valuesequally spaced?First differencesconstant?Second differencesconstant?Make the steps equal(or use rates of change)LINEARQUADRATICNeither linearnor quadraticnoyesyesyesnono
A decision path for any table of values. First make sure the x-steps are equal. Then test the first differences, then the second differences.
Worked example

Example 2: Linear, quadratic or neither?

Classify each relation (x = 0, 1, 2, 3, 4 in every table).

A: y = 3, 7, 11, 15, 19

B: y = 1, 3, 9, 27, 81

C: y = 0, −3, −4, −3, 0

  1. A. First differences: 4, 4, 4, 4. Constant, so linear.
  2. B. First differences: 2, 6, 18, 54. Second differences: 4, 12, 36. Neither is constant, so neither linear nor quadratic. (Each y is 3 times the one before: this is an exponential pattern, which you will meet later in this unit.)
  3. C. First differences: −3, −1, 1, 3. Second differences: 2, 2, 2. Constant and not zero, so quadratic. The y-values also go down and back up symmetrically, which is the shape of a parabola.

A is linear, B is neither (exponential), C is quadratic.

Worked example

Example 3: Real data is never perfect

A ball is tossed upward. A motion sensor records its height h (m) every 0.5 s: 1.5, 5.2, 6.5, 5.3, 1.6. Is a quadratic model reasonable?

  1. The times go up in equal steps of 0.5 s, so differences can be used.
  2. First differences: 5.2 − 1.5 = 3.7, 6.5 − 5.2 = 1.3, 5.3 − 6.5 = −1.2, 1.6 − 5.3 = −3.7.
  3. Second differences: 1.3 − 3.7 = −2.4, −1.2 − 1.3 = −2.5, −3.7 − (−1.2) = −2.5.
  4. The second differences are not exactly equal, because measurements have small errors, but they are all very close to −2.5.

Yes. The second differences are almost constant (and negative, so the parabola opens down), so a quadratic model fits the data well.

Watch out

  • Differences only work if the x-values go up in equal steps. If a table jumps from x = 1 to x = 2 to x = 5, the differences mean nothing.
  • Subtract in the same order every time: next minus this. Mixing the order changes the signs.
  • Be careful with negatives: 3 − (−1) = 4, not 2.
Just for fun

What do you call a table whose first differences keep changing but whose second differences never do?

Fickle on the outside, loyal on the inside. Mathematicians just call it quadratic.

Hands-on activity

Squares, stairs and differences

You need: about 40 small identical objects (coins, buttons, dried beans or building bricks) and paper.

  1. Build squares with side lengths 1, 2, 3, 4 and 5 objects. Count how many objects each square uses. Record side length and total.
  2. Find the first and second differences. Which kind of relation is it?
  3. Now build "staircases": a staircase of size 1 has 1 object; size 2 has 1 + 2 = 3; size 3 has 1 + 2 + 3 = 6, and so on up to size 5. Record the totals and find the differences again.
  4. Compare the two second differences. Which one is bigger, and which pattern grows faster?

What to look for: squares use 1, 4, 9, 16, 25 objects; first differences 3, 5, 7, 9; second differences all 2 (quadratic, y = x²). Staircases use 1, 3, 6, 10, 15; first differences 2, 3, 4, 5; second differences all 1 (also quadratic: y = ½x² + ½x, with a = ½, and 2a = 1). The squares grow faster because their second difference is bigger.

Your turn

1

Find the first and second differences for y = x² + 2 for x = 0, 1, 2, 3, 4.

2

Classify: x = 1, 2, 3, 4, 5 and y = 10, 7, 4, 1, −2.

3

Classify: x = 0, 1, 2, 3, 4 and y = 5, 4, 1, −4, −11.

4

Make a table for y = −3x² for x = −2, −1, 0, 1, 2 and find the second differences. How do they compare with 2a?

5

Why can't you use differences on this table? x = 1, 2, 4, 8 and y = 1, 4, 16, 64.

6

The second differences of a table (x-steps of 1) are all 10. What is the value of a in y = ax² + bx + c? Does the parabola open up or down?

More practice: warm-up

1

Find the first differences: y = 4, 9, 16, 25.

2

Find the second differences from first differences 5, 7, 9.

3

If the first differences are constant, what kind of relation is it?

4

If the second differences are constant and not zero, what kind of relation is it?

5

Calculate y = x² − 1 at x = −2.

More practice: on your own

1

Make a table for y = x² − 4x + 3 for x = 0 to 5 and find the second differences.

2

Classify: x = 0, 1, 2, 3, 4 and y = 2, 4, 8, 16, 32.

3

Fill in the missing value so the table is quadratic: x = 0, 1, 2, 3, 4 and y = 1, 2, 5, 10, ?

4

The number of handshakes when n people all shake hands once with each other is 0, 1, 3, 6, 10 for n = 1, 2, 3, 4, 5. Is this relation quadratic? Predict the number for 6 people.

5

A table has second differences of −0.8 (x-steps of 1). Find a and describe the parabola.

Challenge

1

Show with algebra that for y = ax² + bx + c the second difference is 2a when x goes up by 1. (Hint: find y at x, x + 1 and x + 2.)

2

For y = x², use x = 0, 2, 4, 6, 8 (steps of 2). Find the second differences. Why are they not equal to 2?

3

A quadratic table has y = 3 at x = 0, first difference 5 between x = 0 and x = 1, and constant second difference −2. Find y at x = 1, 2, 3 and 4.

Check your understanding

Quiz

1.

The second differences of a table are all 6 (x-steps of 1). The relation is…

2.

Find the first differences of y = 1, 4, 9, 16, 25.

3.

Which table is quadratic? (x = 0, 1, 2, 3 in each)

4.

For y = 4x² − x + 7 with x-steps of 1, the second difference is…

5.

Before using differences, you must check that…

6.

A table has first differences 2, 6, 18, 54. It is…

For parents and teachers

Big idea: Expectation 1.2 asks students to find, with and without technology, that y = ax² + bx + c (a ≠ 0) has a constant second difference in its table of values and graphs as a parabola. This lesson focuses on the tables; the next lesson focuses on the graphs.

How to help: Ask your teen to explain the difference between first and second differences in their own words. The bean or coin squares make a good kitchen-table demonstration. A spreadsheet is a quick way to check: put x in column A, the y formula in column B, and =B2-B1 style formulas in the next columns.

Common difficulties:

  • Arithmetic with negatives, especially subtracting a negative. Encourage writing the subtraction out: 3 − (−1).
  • Using differences when x is not evenly spaced. Ask them to check the x-column first, every time.
  • Mixing up the order (this − next instead of next − this), which flips the signs.
  • Expecting real data to give perfectly equal second differences. "Close to constant" is good enough for measured data.