Math worksheet with problems on linear and non-linear functions, graphing, and coordinate relationships.

Let’s go step by step through each question.

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Question 6: Fill in the table for y = x² + 2

We plug in each x value into the equation to find y.

- When x = -2 → y = (-2)² + 2 = 4 + 2 = 6
- When x = -1 → y = (-1)² + 2 = 1 + 2 = 3
- When x = 0 → y = (0)² + 2 = 0 + 2 = 2
- When x = 1 → y = (1)² + 2 = 1 + 2 = 3
- When x = 2 → y = (2)² + 2 = 4 + 2 = 6

So the completed table is:

| x | y |
|----|----|
| -2 | 6 |
| -1 | 3 |
| 0 | 2 |
| 1 | 3 |
| 2 | 6 |

Now, plot these points on the coordinate plane:
(-2,6), (-1,3), (0,2), (1,3), (2,6)

You’ll see they form a U-shape — that’s called a parabola.

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Question 7: Describe the relationship between coordinates and graph

As x gets farther from 0 (either positive or negative), y increases. The graph is symmetric — meaning if you fold it down the middle at x=0, both sides match. It’s not a straight line — it curves upward.

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Question 8: Is this linear? What looks different? Why can’t it be a straight line?

No, this is not linear.

What looks different: The graph curves — it doesn’t go in one straight direction. Also, when we look at how y changes as x increases, it doesn’t change by the same amount every time.

For example:
- From x=-2 to x=-1: y goes from 6 to 3 → change of -3
- From x=-1 to x=0: y goes from 3 to 2 → change of -1
- From x=0 to x=1: y goes from 2 to 3 → change of +1
- From x=1 to x=2: y goes from 3 to 6 → change of +3

The rate of change isn’t constant — which means it can’t be a straight line. Linear equations have a constant rate of change (same slope everywhere). This one has changing slopes — so it curves.

Also, the equation has an x² term — any equation with x squared will make a curve, never a straight line.

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Question 9: Fill in the table for y = |x+2|

Absolute value means “distance from zero” — always positive or zero.

Let’s calculate:

- x = -2 → y = |-2 + 2| = |0| = 0
- x = -1 → y = |-1 + 2| = |1| = 1
- x = 0 → y = |0 + 2| = |2| = 2
- x = 1 → y = |1 + 2| = |3| = 3
- x = 2 → y = |2 + 2| = |4| = 4

Completed table:

| x | y |
|----|----|
| -2 | 0 |
| -1 | 1 |
| 0 | 2 |
| 1 | 3 |
| 2 | 4 |

Plot these points: (-2,0), (-1,1), (0,2), (1,3), (2,4)

Wait — actually, let’s double-check! Absolute value graphs usually have a V shape. But here, since we’re adding inside the absolute value, the vertex shifts.

Actually, let’s test x = -3 just to see the full picture (even though not asked):

x = -3 → y = |-3+2| = |-1| = 1 → point (-3,1)

But our table only asks for x from -2 to 2.

Looking at our calculated values: from x=-2 to x=2, y keeps increasing: 0,1,2,3,4 — that’s a straight line going up!

Wait — that seems odd. Let me check again.

Hold on — maybe I made a mistake.

The function is y = |x + 2|

At x = -2 → |0| = 0 ✔
At x = -1 → |1| = 1 ✔
At x = 0 → |2| = 2 ✔
At x = 1 → |3| = 3 ✔
At x = 2 → |4| = 4 ✔

So yes — for x ≥ -2, the expression inside the absolute value is non-negative, so |x+2| = x+2 — which IS linear!

But what about x < -2? For example, x = -3 → |-1| = 1, but x+2 = -1 → so |x+2| ≠ x+2 there.

However, in our table, all x values are ≥ -2, so in this range, the graph *is* a straight line.

But wait — the problem says “this relationship is not linear” in question 11. That suggests maybe I misread the function?

Let me re-read Question 9: “y = |x+2|”

And Question 11 says: “This relationship is not linear.”

Hmm — contradiction?

Wait — perhaps the student is supposed to realize that even though in this limited domain it looks linear, the FULL graph of y = |x+2| is NOT linear — because if you include x < -2, it bends.

For example:

x = -4 → y = |-4+2| = |-2| = 2
x = -3 → y = 1
x = -2 → y = 0
x = -1 → y = 1
x = 0 → y = 2

So plotting more points: (-4,2), (-3,1), (-2,0), (-1,1), (0,2) — now you see the V shape!

Ah — so even though the table only shows x from -2 to 2 (which gives a straight line segment), the actual function over all real numbers is NOT linear — it has a corner at x = -2.

So for Question 9, based on the given x-values, the y-values are correct as above.

But for Questions 10 and 11, we must consider the whole function.

So let’s proceed.

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Question 10: Describe the relationship between coordinates and graph

If we plot only the points from the table (-2,0) to (2,4), they lie on a straight line with slope 1.

BUT — if we consider the entire function y = |x+2|, the graph is V-shaped, with a sharp turn at x = -2. To the left of x=-2, the line goes downward; to the right, it goes upward.

So the full graph is two straight lines meeting at a point — not one single straight line.

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Question 11: Is this linear? What looks different? Why can’t it be a straight line?

Technically, the function y = |x+2| is piecewise linear, but overall, it is not a linear function because its graph is not a single straight line — it has a corner (vertex) at x = -2.

What looks different: If you were to extend the graph to the left (x < -2), you’d see the line turns — it doesn’t keep going in the same direction.

Why it can’t be a straight line: A true linear function must have the same slope everywhere. Here, for x < -2, the slope is -1 (because y = -(x+2)), and for x > -2, the slope is +1 (because y = x+2). Since the slope changes, it’s not linear.

Even though the table only shows the right side (where it looks linear), the function itself is defined for all x, and globally, it’s not linear.

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Question 12: Summarize 3 facts learned about linear functions

1. Linear functions have equations like y = mx + b — no exponents, no absolute values, no squares.
2. Their graphs are always straight lines — no curves, no corners.
3. They have a constant rate of change — meaning for every equal step in x, y changes by the same amount.

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Final Answer:

Question 6 Table:

| x | y |
|----|----|
| -2 | 6 |
| -1 | 3 |
| 0 | 2 |
| 1 | 3 |
| 2 | 6 |

Question 7: The graph is curved (U-shaped), symmetric around x=0, and y increases as x moves away from 0.

Question 8: Not linear. Graph curves. Rate of change isn't constant. Equation has x² — causes curvature.

Question 9 Table:

| x | y |
|----|----|
| -2 | 0 |
| -1 | 1 |
| 0 | 2 |
| 1 | 3 |
| 2 | 4 |

Question 10: Points from table lie on a straight line, but full graph of y=|x+2| is V-shaped with a corner at x=-2.

Question 11: Not linear. Full graph has a corner — slope changes from -1 to +1 at x=-2. Linear functions must have constant slope.

Question 12 Summary:
1. Linear functions have straight-line graphs.
2. Their equations are in the form y = mx + b (no powers or absolute values).
3. They have a constant rate of change (slope).