Increasing, Decreasing, Linear and Nonlinear Intervals | Grade 8 Math | LevelKite

A function's graph pairs each input $x$ with exactly one output $y$ . In this lesson, $x$ and $y$ are plain numbers, each grid square is 1 unit on both axes, and every graph is one connected path drawn for the inputs from $x = 0$ to $x = 10$ . That set of inputs is the path's domain.

You read a graph from left to right, the direction in which $x$ increases. Over an interval, a stretch of inputs from $x = a$ to $x = b$ where $b$ is greater than $a$ , a function is:

increasing when the outputs go up as $x$ goes up,
decreasing when the outputs go down as $x$ goes up,
constant when the outputs stay the same, which shows as a flat part.

A turning point is a point where the graph switches from increasing to decreasing or from decreasing to increasing. One interval ends and the next begins at each turning point and at each end of a flat part. Always name an interval by its inputs, never by its heights (outputs) alone.

Straight pieces join

(0, 1)

,

(3, 4)

,

(5, 4)

,

(7, 1)

and

(10, 4)

; the dots mark the points at

x = 3

,

x = 5

and

x = 7

.

Worked example

Direction on a path of straight pieces

The graph above is made of straight pieces joining $(0, 1)$ , $(3, 4)$ , $(5, 4)$ , $(7, 1)$ and $(10, 4)$ , in that order. Name each interval where the function is increasing, decreasing or constant, and name any turning point.

Read from left to right. The inputs where the pieces meet split the domain into four intervals, and you will name each interval by these inputs.
$0 < 3 < 5 < 7 < 10$
From x = 0 to x = 3, the output goes from 1 to 4. A straight piece goes only one way, so comparing its two ends is enough. You are moving right, so a positive change in y means the outputs go up as x goes up: the function is increasing.
$4 - 1 = 3 > 0$
From x = 3 to x = 5, both ends have output 4, so the change is 0. This flat part is constant, and its ends, x = 3 and x = 5, are where the description changes.
$4 - 4 = 0$
From x = 5 to x = 7, the output goes from 4 down to 1. The change is negative, so the function is decreasing.
$1 - 4 = -3 < 0$
From x = 7 to x = 10, the output goes from 1 back up to 4, so the function is increasing. It decreases just before x = 7 and increases just after, so (7, 1) is a turning point.
$4 - 1 = 3 > 0$

Answer

The function is increasing from $x = 0$ to $x = 3$ , constant from $x = 3$ to $x = 5$ , decreasing from $x = 5$ to $x = 7$ , and increasing from $x = 7$ to $x = 10$ , with a turning point at $(7, 1)$ .

Next, look at the shape of each piece. You know that a line has one slope, so its output changes by the same amount for every step of 1 in $x$ . An interval is linear when the graph over it is one straight piece with one slope. A flat piece is linear too: its slope is 0.

The curved pieces in this lesson are parts of parabolas, graphs of equations with an $x^2$ term. A curved piece has no single slope. Its steepness, how much $y$ rises or falls for each step of 1 in $x$ , keeps changing, so the piece gets steeper or less steep as you move right. An interval where the graph curves is nonlinear.

Check with coordinates, not with how the picture looks: use the equation to find exact outputs at equal steps of $x$ . If the changes in $y$ are not all equal, the piece cannot be straight, so it is nonlinear. Compare the pieces of one graph this way, and name each linear or nonlinear interval by its inputs, from $x = a$ to $x = b$ .

Straight from

(0, 0)

to

(2, 4)

, curved along

y = -0.25x^2 + 3x - 1

up to

(6, 8)

, then flat to

(10, 8)

; dots mark

x = 2

and

x = 6

.

Worked example

Linear or nonlinear?

The graph above has three pieces: a straight piece from $(0, 0)$ to $(2, 4)$ , a curved piece that follows $y = -0.25x^2 + 3x - 1$ from $(2, 4)$ to $(6, 8)$ , and a flat piece from $(6, 8)$ to $(10, 8)$ . Name each interval as linear or nonlinear, and describe how the steepness of the curved piece changes.

Start at the left with the straight piece and find its slope from its endpoints. The change in x, 2 − 0 = 2, is not zero, so the slope is defined. One slope means the interval from x = 0 to x = 2 is linear: every step of 1 in x raises y by 2.
$\text{slope} = \frac{4 - 0}{2 - 0} = \frac{4}{2} = 2$
For the curved piece, substitute x = 3 into its equation. The output is exact because it comes from the equation, not from estimating on the grid.
$y = -0.25(3)^2 + 3(3) - 1 = -2.25 + 9 - 1 = 5.75$
Substitute x = 4 and x = 5 the same way, so you have outputs at equal steps of 1 across the curve. Here −0.25 × 16 = −4 and −0.25 × 25 = −6.25.
$x = 4:\quad y = -4 + 12 - 1 = 7,\qquad x = 5:\quad y = -6.25 + 15 - 1 = 7.75$
Subtract each output from the next, using the given ends (2, 4) and (6, 8). The rises shrink from 1.75 to 0.25, while the straight piece rose 2 for every step. Unequal changes mean the curve has no single slope: it gets less steep as x increases, so the interval from x = 2 to x = 6 is nonlinear.
$5.75 - 4 = 1.75,\quad 7 - 5.75 = 1.25,\quad 7.75 - 7 = 0.75,\quad 8 - 7.75 = 0.25$
The flat piece keeps the output at 8 from x = 6 to x = 10, so its slope is 0. One slope, even 0, makes this interval linear.
$\frac{8 - 8}{10 - 6} = \frac{0}{4} = 0$

Answer

The graph is linear from $x = 0$ to $x = 2$ with slope 2, nonlinear from $x = 2$ to $x = 6$ , where it gets less steep, and linear from $x = 6$ to $x = 10$ with slope 0.

Key concept

Two questions for every interval

Read the graph from left to right and name every interval by its inputs, from $x = a$ to $x = b$ .

Direction: Do the outputs go up (increasing), go down (decreasing) or stay level (constant, a flat part)? A direction interval ends at a turning point or where a flat part begins or ends.
Shape: Compare the pieces of the same graph. Is the interval one straight piece with one slope (linear), or a curved piece whose steepness keeps changing, getting steeper or less steep (nonlinear)?

The two questions can split the inputs at different places, so answer each one on its own.

Straight from

(0, 6)

to

(2, 4)

, curved along

y = 0.5x^2 - 4x + 10

through the turning point

(4, 2)

to

(6, 4)

, then straight to

(10, 8)

.

Worked example

Direction and shape together

The graph above is straight from $(0, 6)$ to $(2, 4)$ , follows $y = 0.5x^2 - 4x + 10$ from $(2, 4)$ to $(6, 4)$ , and is straight from $(6, 4)$ to $(10, 8)$ . The curved piece turns at its lowest point, $(4, 2)$ . Describe where the function is increasing or decreasing and where it is linear or nonlinear.

Check the given turning point with the equation, not the picture. The outputs at x = 3, 4 and 5 are 2.5, 2 and 2.5, so the curve comes down to 2 at x = 4 and goes back up. That agrees with (4, 2) being its lowest point, and x = 4 is inside the curve's domain, the inputs from x = 2 to x = 6.
$x = 3:\quad 4.5 - 12 + 10 = 2.5,\qquad x = 4:\quad 8 - 16 + 10 = 2,\qquad x = 5:\quad 12.5 - 20 + 10 = 2.5$
Notice that the curve's ends, (2, 4) and (6, 4), have the same height, yet the curve is not flat: it dips to 2 between them. You can judge a straight piece by its ends, but a curved piece can turn in between, so its turning point decides where its direction changes.
$\text{ends: } 4 - 4 = 0,\qquad \text{lowest output: } 2$
Read direction from left to right. The straight piece falls from 6 to 4 and the curve keeps falling to its turning point, so the function is decreasing from x = 0 to x = 4. Then the curve rises to 4 and the last straight piece keeps rising to 8, so the function is increasing from x = 4 to x = 10.
$6 \to 4 \to 2 \text{ for } x = 0 \text{ to } 4,\qquad 2 \to 4 \to 8 \text{ for } x = 4 \text{ to } 10$
Now look at shape. Each straight piece has one slope, found from its endpoints, so the intervals from x = 0 to x = 2 and from x = 6 to x = 10 are linear.
$\frac{4 - 6}{2 - 0} = \frac{-2}{2} = -1,\qquad \frac{8 - 4}{10 - 6} = \frac{4}{4} = 1$
On the curve, equal steps of 1 change y by −1.5, −0.5, 0.5 and 1.5. These are not equal: the curve gets less steep on the way down and steeper on the way up, so the interval from x = 2 to x = 6 is nonlinear. The shape splits the inputs at x = 2 and x = 6, but the direction splits them at x = 4.
$2.5 - 4 = -1.5,\quad 2 - 2.5 = -0.5,\quad 2.5 - 2 = 0.5,\quad 4 - 2.5 = 1.5$

Answer

The function is decreasing from $x = 0$ to $x = 4$ and increasing from $x = 4$ to $x = 10$ , turning at $(4, 2)$ . It is linear from $x = 0$ to $x = 2$ and from $x = 6$ to $x = 10$ , and nonlinear from $x = 2$ to $x = 6$ .

Takeaway: Read a graph from left to right and name each interval by its inputs, from $x = a$ to $x = b$ , never by heights alone. On each interval the function is increasing, decreasing or constant, and its direction changes at turning points and at the ends of flat parts. Compare the pieces of one graph: an interval is linear where one straight piece keeps one slope, and nonlinear where a curve keeps getting steeper or less steep. Next, you will use these descriptions to interpret graphs of real-world situations, sketch a graph from a description in words, and find and correct errors in reading graphs.