Reading values off a graph
You have the graph of a function but not its rule. How do you get the output for a given input, or every input that gives a certain output, from the picture alone?
Where you are 5 in Graphs of functions
When you reach for it
Often a graph is all you have: a printout from a heart monitor, a chart of prices, a curve in a report. There is no rule to put numbers into, yet the graph holds every pair of input and output the function has. This lesson reads those pairs off the picture.
Two questions come up, and they go in opposite directions.
- Given an input, what is the output? In function notation: find f(3).
- Given an output, which inputs give it? In function notation: find every x with f(x) = 2.
The first always has at most one answer. The second can have none, one, two or many, and finding all of them is most of the work.
When there is a rule, putting the number into the rule gives an exact output. Reading a graph is for when the picture is what you have, or when you want to see every answer at once. Saying in words what a graph’s shape tells you happened is Graphs that tell a story; here we read numbers.
The procedure
What a point on the graph says
The graph of a function f is the set of all points (x, y) with y = f(x): each input paired with its own output. So the two statements
say exactly the same thing, one in Function notation and one as a picture. Every reading in this lesson translates one into the other. The point (4, 3) on the graph of f is the statement f(4) = 3; the statement f(-2) = 0 is the point (-2, 0).
From an input to its output
To find f(a), find a on the horizontal axis, which carries the inputs. Go straight up or down from there until you meet the graph. Then go straight across to the vertical axis, which carries the outputs, and read the number there. That number is f(a).
Why this works: moving straight up or down keeps the first coordinate equal to a, so the point where you meet the graph is the one point of the graph whose input is a, namely (a, f(a)). Moving straight across keeps its height, and its height is its output. Figure 1 shows the move for f(4): up from 4, meet the line at (4, 3), across to 3. So f(4) = 3.
Where the graph lies below the horizontal axis, you go down to it, and the output is negative.
You meet the graph at one point at most, because a function gives each input exactly one output; that is what the Testing graphs and equations for functions checks. If the vertical line through a misses the graph altogether, a is not an input of the function and f(a) has no value.
From an output to every input
To solve f(x) = b, find b on the vertical axis and draw, or imagine, the horizontal line at height b. Each point where that line meets the graph has output b. From each one, go straight down or up to the horizontal axis and read its input. The solutions are all of those inputs.
“All” is the difference from the first question. A function gives each input only one output, but nothing stops several inputs from sharing an output. A horizontal line can miss the graph (no solution), touch it once, cross it twice, meet it three or more times, or, on a graph with a flat piece, run along the whole piece, so that every input under it is a solution. Sweep the line across the whole width of the graph rather than stopping at the first meeting.
Reading between the grid lines
A graph is a drawing. When the point you need sits exactly on a grid corner, where two grid lines cross, read its coordinates straight off. When it falls between grid lines, judge the fraction of a square (“about halfway”, “a little more than a quarter”) and say plainly that the answer is an estimate: write “about 10.5”, not “10.5”. A finer grid gives a better estimate, but still an estimate.
One case does better than an estimate. Where a straight piece of graph joins two grid corners, the point halfway across is exactly halfway up, because a straight line rises by equal amounts over equal distances across. A curve makes no such promise.
Read the scales first
Before reading anything, check what one square stands for on each axis. Graphs of real quantities seldom use 1 per square, the two axes usually differ, and an axis need not start at 0: in Figure 2 the temperature axis starts at 8.
Endpoints. Where a graph stops, a filled dot means the end point belongs to the graph and an open circle means it does not; Graphing a piecewise function uses both.
Worked examples
Reading outputs off a line
Figure 1 is the graph of a function f, from x = -4 to x = 6 with both end points included. Find (a) f(-2), (b) f(0), (c) f(-4) and (d) f(3).
Solution(a) The line crosses the horizontal axis exactly at -2, so the point of the graph above -2 is (-2, 0) and no move up or down is needed. Its height is 0, so f(-2) = 0.
(b) Going up from 0 means going up the vertical axis itself, and the line meets it at the grid corner (0, 1). So f(0) = 1. The output for input 0 is always the height where a graph crosses the vertical axis.
(c) At -4 the line is below the horizontal axis, so go down. It ends at the filled dot on the grid corner (-4, -1), so f(-4) = -1. A point below the axis has a negative output.
(d) Going up from 3 meets the line between grid lines, between heights 2 and 3. Here we can be exact. The line passes through the grid corners (2, 2) and (4, 3), and 3 is halfway from 2 to 4; on a straight line, halfway across is halfway up, so the height is halfway from 2 to 3:
Every input for a given output
Using Figure 1 again, solve (a) f(x) = 2, (b) f(x) = -1 and (c) f(x) = 5.
Solution(a) The horizontal line at height 2 meets the graph once, at the grid corner (2, 2). Going down from there to the horizontal axis gives x = 2. There is only one meeting because the line rises the whole way, so it reaches each height once. The solution is x = 2, and indeed f(2) = 2.
(b) The horizontal line at height -1 meets the graph only at its left end, (-4, -1). The solution is x = -4.
(c) Height 5 is the top edge of the grid, and the drawn graph never gets there: it stops at the filled dot (6, 4). For the function as drawn, with inputs from -4 to 6, the equation f(x) = 5 has no solution. Only if we assume the line carries on past x = 6, rising 1 for every 2 across, would it reach height 5, at x = 8. That is a guess about a part nobody drew, not a reading, and must be labeled as one.
A day’s temperature at a weather buoy
Figure 2 shows T(h), the air temperature in degrees Celsius at a weather buoy h hours after midnight, over one day. Find (a) T(12), (b) T(16) and (c) T(2), and say what T(12) means.
First the scales. Across, each grid line is 1 hour and the labels run every 4 hours. Up, each grid line is 1^\circ\text{C}, and the scale starts at 8.
(a) Go up from 12 to the curve. It meets the curve at a grid corner, level with 16. So T(12) = 16: at noon the temperature at the buoy was 16^\circ\text{C}.
(b) Up from 16 the curve is at its highest point, again on a grid corner, level with 18. So T(16) = 18: at 4 pm it was 18^\circ\text{C}.
(c) Up from 2 the curve is between the grid lines for 10 and 11, a little above halfway. So T(2) is about 10.5: about 10.5^\circ\text{C} at 2 am. The grid cannot settle the tenth.
Every time the temperature took a value
Using Figure 2, find every time when the temperature was (a) 14^\circ\text{C}, (b) 18^\circ\text{C}, (c) 19^\circ\text{C} and (d) 11^\circ\text{C}.
SolutionEach part asks for every solution of T(h) = b for one height b, so each is a horizontal line swept across the whole day.
(a) The dashed line at 14 meets the curve twice, at the grid corners (10, 14) and (22, 14). Going down from each: h = 10 and h = 22. It was 14^\circ\text{C} at 10 am, while the temperature was rising, and again at 10 pm, while it was falling.
(b) The line at 18 touches the curve only at its highest point, (16, 18), and nowhere else. One solution: h = 16, at 4 pm.
(c) The line at 19 is above the highest point of the curve and never meets it. There is no solution: the temperature never reached 19^\circ\text{C} that day.
(d) The line at 11 crosses the curve twice, and both crossings are between grid lines: one between h = 1 and h = 1.5, the other between h = 6.5 and h = 7. So the solutions are about h = 1.2 and about h = 6.8: a little after 1 am, on the way down to the night’s low, and a little before 7 am, on the way back up.
A step graph: postage on a letter
Figure 3 shows C(w), the cost in dollars of posting a letter that weighs w grams, for letters up to 100 g. Find (a) C(35) and (b) C(20). (c) Which weights cost 1.75 dollars? (d) Does any letter cost 2 dollars?
(a) Go up from 35. The first flat piece ends at 20, so the line up from 35 meets the middle piece, at height 1.75. So C(35) = 1.75: a 35 g letter costs 1.75 dollars.
(b) At 20 two pieces end: the lower one with a filled dot at (20, 1.25) and the middle one with an open circle at (20, 1.75). The filled dot is on the graph and the open circle is not, so the only point of the graph above 20 is (20, 1.25), and C(20) = 1.25. A letter of exactly 20 g still pays the lower price.
(c) The horizontal line at 1.75 runs along the whole middle piece, from the open circle at 20 to the filled dot at 50. Every weight under that piece is a solution: every w with
So 20.1 g, 37 g and 50 g all cost 1.75 dollars, and 20 g does not.
(d) The line at 2 passes between the middle piece, at 1.75, and the top piece, at 2.50, meeting neither. No letter costs 2 dollars.
Using readings: the warm part of the day
Using Figure 2, (a) by how much did the temperature rise from 4 am to 4 pm, and (b) for how long was it at least 16^\circ\text{C}?
Solution(a) Read two outputs and subtract. At h = 4 the curve is at its lowest point, the grid corner (4, 10), so T(4) = 10. From Example 3, T(16) = 18. The rise is
degrees Celsius.
(b) “At least 16” means T(h) \ge 16: the times when the curve is on or above the horizontal line at 16. First find where the curve meets that line: at the grid corners (12, 16) and (20, 16), so h = 12 and h = 20. Between those times the curve is above the line, and before 12 and after 20 it is below. So T(h) \ge 16 exactly when 12 \le h \le 20, from noon to 8 pm, which is 20 - 12 = 8 hours.
A question with “at least” or “less than” is read the same way: find where the graph meets the line, then see which side of the line the graph is on between those points.
Boundaries
Only what is drawn. A graph shows a window: some inputs, some outputs. A value outside the window is not in the picture (Example 2(c)), and continuing a line or curve past its edge is an assumption about the function, not a reading.
Readings are estimates unless they land on a grid corner, or halfway along a straight piece between corners. When an exact value is needed and there is a rule, the rule gives it.
An input with no point above or below it has no output. If the vertical line through a misses the graph, f(a) is undefined, not 0; Domain and range from a graph collects the inputs that do have points.
The curve must be the graph of a function. A curve that some vertical line meets twice, such as a circle, does not give one output per input, and f(a) means nothing for it.
Where this goes next
The landmarks met here, where a graph crosses the axes and where it is highest and lowest, get names in Features of a graph. Collecting every input that has an output and every output that is reached gives Domain and range. Solving f(x) = g(x) with graphs and tables draws two graphs on the same axes and reads off the inputs where they meet, the horizontal-line reading with a second graph in place of the line. Evaluating a composition from numbers, tables and graphs reads an output off one graph and feeds it into another as an input, and Adding, subtracting and multiplying functions reads two outputs at the same input and adds them. Graphs that tell a story turns from numbers to the story a graph’s shape tells.
Exercises
Reading outputs
Figure 4 is the graph of a function h, drawn from x = -2 to x = 4 with both end points included. Find each value.
- h(0)
- h(3)
- h(1)
Every input for an output
Use Figure 4. Find every solution, or say there is none. Where a point is between grid lines, give an estimate.
- h(x) = 3
- h(x) = 4
- h(x) = 5
- h(x) = -5
- h(x) = 2
The weather buoy
Use Figure 2.
- Find T(4) and T(20), and say in a sentence what each means.
- At what times of the day was the temperature 12^\circ\text{C}?
- At what times was it 17^\circ\text{C}? Give estimates.
The reservoir
Figure 5 shows D(m), the depth of water in meters at a reservoir’s dam m months after 1 January, over one year.
- Find D(2) and D(3).
- When was the depth 16 meters?
- When was the depth 18 meters?
- When was the depth 10 meters?
Postage
Use Figure 3.
- Imogen-Lee posts a letter weighing 50 g, and Jabari one weighing 51 g. What does each pay?
- Guadalupe paid 1.25 dollars to post a letter. What could it weigh?
Reasoning
- Explain why reading f(a) off the graph of a function gives at most one answer, while solving f(x) = b off the same graph can give several.
- Eithne reads T(14) off Figure 2 as 17.5, and Florentin reads it as 17.4. Can both be right? Say what the graph can tell you about T(14) and what it cannot.
- The graph of a function f passes through (2, 5) and (7, 5), and through no other point at height 5. Write both facts in function notation, solve f(x) = 5, and explain why f(4) cannot be 5.
Answers
- h(0) = 3. Going up the vertical axis, the curve crosses it at the grid corner (0, 3).
- h(3) = 0. The curve crosses the horizontal axis at (3, 0), so the height there is 0.
- h(1) = 4. Up from 1 the curve is at its highest point, the grid corner (1, 4).
- x = 0 and x = 2. The horizontal line at 3 crosses the curve twice, at (0, 3) and (2, 3).
- x = 1 only. The line at 4 touches the curve at its highest point, (1, 4), and nowhere else.
- No solution. The line at 5 is above the highest point of the curve, which is at height 4.
- x = -2 and x = 4. The line at -5 meets the graph only at its two end points, and the filled dots say both belong to it.
- About x = -0.4 and about x = 2.4. The line at 2 crosses the curve between -1 and 0, a little less than halfway from 0 toward -1, and between 2 and 3, a little less than halfway from 2 toward 3. Both are estimates.
- T(4) = 10: at 4 am the temperature was 10^\circ\text{C}, the lowest of the day. T(20) = 16: at 8 pm it was 16^\circ\text{C}. Both points are grid corners.
- At h = 0, h = 8 and h = 24: midnight at the start of the day, 8 am, and midnight at the end. The line at 12 meets the curve three times, at the grid corners (0, 12), (8, 12) and (24, 12); a horizontal line can meet a curve more than twice.
- About h = 13.2 and about h = 18.8, a little after 1 pm and a little before 7 pm. The line at 17 crosses the curve between h = 13 and h = 13.5, and again between h = 18.5 and h = 19.
- D(2) = 16, a grid corner. D(3) = 17: 3 is halfway along the straight piece from (2, 16) to (4, 18), so the depth is halfway from 16 to 18.
- At m = 2 and m = 7. The line at 16 meets the graph at (2, 16), and again halfway along the straight piece from (6, 18) to (8, 14), which falls from 18 to 14, at (7, 16).
- At every time from m = 4 to m = 6, that is 4 \le m \le 6. The graph is flat at 18 between (4, 18) and (6, 18), so the line at 18 runs along the whole piece.
- At about m = 9.3 and at m = 12. The straight piece from (8, 14) to (10, 8) falls 6 meters in 2 months; 10 is 4 meters below 14, which is two thirds of the way down, so two thirds of the way across, at \displaystyle 8 + \frac{4}{3}, about 9.3. The graph also ends at the filled dot (12, 10).
- Imogen-Lee pays 1.75 dollars: 50 g carries the filled dot at the right end of the middle piece. Jabari pays 2.50 dollars: 51 g is past 50, under the top piece.
- Any weight greater than 0 g and at most 20 g, that is 0 < w \le 20. The line at 1.25 runs along the whole lowest piece, from the open circle at 0 to the filled dot at 20.
- Reading f(a) means finding the point of the graph straight above or below a. A function gives each input exactly one output, so there is at most one such point. Solving f(x) = b means finding every point at height b, and nothing stops different inputs from sharing an output: in Figure 4, h(0) = 3 and h(2) = 3.
- Yes, both fit. Up from 14 the curve sits between the grid lines for 17 and 18, close to halfway, so the graph says T(14) is about 17.5. It cannot say which tenth is right: the grid is drawn every whole degree and the point is not on a corner.
- f(2) = 5 and f(7) = 5. The solutions of f(x) = 5 are x = 2 and x = 7, since these are the only points of the graph at height 5. If f(4) were 5, the point (4, 5) would be on the graph, a third point at height 5, which the question rules out.