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How to read a graph of a function: values, domain, range, intercepts

The rule: every question is a direction to look in

A graph question is never about calculation - it is about which way you move your finger. A value $f (a)$ : find $a$ on the horizontal axis, go up or down to the curve, read the height. The domain: the shadow of the whole curve on the horizontal axis; the range: its shadow on the vertical one. Intercepts: the $x$ -intercepts are where the curve meets the horizontal axis (there may be several), the $y$ -intercept is where it meets the vertical one - and a function has at most one. Where $f (x) > 0$ : the stretches of $x$ where the curve runs above the axis; the answer is intervals of $x$ , never of $y$ . Solving $f (x) = c$ : draw the horizontal line at height $c$ and read off every $x$ where it touches the curve. The vertical-line test: any vertical line may cross the graph at most once - that is what makes it a function.

Problem

The graph of $y = f (x)$ is the polyline through the points $(- 14, - 8)$ , $(- 12, 0)$ , $(- 6, 12)$ , $(- 4, 8)$ , $(- 2, 0)$ , $(0, - 4)$ , $(4, - 8)$ , $(8, 0)$ , $(12, 8)$ . Answer the usual set of questions: (a) Find $f (- 14)$ and $f (- 6)$ . (b) State the domain and the range. (c) List the $x$ -intercepts and the $y$ -intercept. (d) For what values of $x$ is $f (x) > 0$ ? (e) For what $x$ is $f (x) = - 8$ ? And $f (x) = 12$ ? (f) How often do the lines $y = 1$ and $x = 2$ meet the graph?

The graph. Reading a value, an intercept or a sign is always the same move: pick a direction and follow it to the curve.

Answer

f (- 14) = - 8

f (- 6) = 12

. Domain

[- 14, 12]

, range

[- 8, 12]

x

-intercepts:

- 12, - 2, 8

;

y

-intercept:

- 4

f (x) > 0

(- 12, - 2) \cup (8, 12]

f (x) = - 8

x = - 14

and

x = 4

;

f (x) = 12

x = - 6

. The line

y = 1

crosses three times, the line

x = 2

exactly once.

Checked by hand: every answer was read off the graph twice.

Step-by-step solution

(a) Values of the function

For $f (- 14)$ go to $x = - 14$ - the left end of the graph - and read the height there: the point is $(- 14, - 8)$ , so

f (- 14) = - 8

The same move at $x = - 6$ lands on the highest point of the graph:

f (- 6) = 12

(b) Domain and range

The curve starts at $x = - 14$ and ends at $x = 12$ , with no gaps between, so the domain is that whole stretch:

D : [- 14, 12]

Heights run from the lowest points $(- 14, - 8)$ and $(4, - 8)$ up to the peak $(- 6, 12)$ :

R : [- 8, 12]

(c) Intercepts

The graph meets the $x$ -axis at three marked points:

x = - 12, x = - 2, x = 8

It crosses the $y$ -axis once, at the point $(0, - 4)$ , so the $y$ -intercept is $- 4$ . A function can never have two: two heights above $x = 0$ would break the vertical-line test.

(d) Where the function is positive

$f (x) > 0$ means the curve is above the axis. Between $x = - 12$ and $x = - 2$ it rises to the peak and comes back down - all above. It then stays below until $x = 8$ , and from there climbs again to the right end:

(- 12, - 2) \cup (8, 12]

The ends $- 12$ , $- 2$ and $8$ are excluded because there $f (x) = 0$ , not more than $0$ ; the end $12$ is included because $f (12) = 8 > 0$ .

(e) Solving f(x) = c from the picture

Draw the horizontal line at the given height and see where it touches. At height $- 8$ it touches twice - at the very left end and at the bottom point on the right:

x = - 14 and x = 4

At height $12$ it touches only the peak:

x = - 6

(f) Counting crossings

The line $y = 1$ is horizontal and slightly above the axis. The curve passes that height on the way up to the peak, on the way down from it, and once more on the final climb - three times. The line $x = 2$ is vertical, and any vertical line inside the domain meets the graph exactly once. That is not a property of this picture but of every function - it is the vertical-line test.

How to check yourself

Answers to (d) and (e) must be consistent: the sign changes exactly at the $x$ -intercepts found in (c). If your positivity interval starts somewhere the curve does not cross the axis, one of the two answers is wrong.

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