Read a graph and choose a method

Explain it like I am five

Think of a graph as a road map. A marked point tells you where the road passes. The tilt tells you how far it rises or falls while you move right. To describe a straight road with an equation, you need its tilt and one place it passes. That place can be where it crosses the vertical axis, called the y-intercept, or another marked point. Read the numbered axes before you count. A square on the picture might represent two units instead of one. If the crossing is outside the picture, use a visible point to work backward to it.

Read the starting output and the output change per input step from this original graph.

Reminder

Ordered pairs. In (4, 8), the first number is input x = 4 and the second is output y = 8.
Solving for a missing constant. To find b in 8 = −6 + b, add 6 on both sides: b = 14. Check: −6 + 14 = 8.
Slope. Slope is output change over input change: $2 − 1 over 1 − 0$ = 1.
Subtracting negatives. 1 − (−3) = 4, so keep parentheses around negative coordinates.
Fractions and signs. $−6 over 4$ reduces to − $3 over 2$ , and − $3 over 2$ × 4 = −6.
Initial value. At input zero, y = mx + b becomes y = b.
Distribution. 2(x − 4) = 2x − 8 because 2 multiplies both terms.

Why it works. A straight line keeps the same rise divided by run between every pair of different inputs. Two readable points therefore determine its slope. Its y-intercept tells you the output at input zero, so together they determine y = mx + b. You do not need to see the intercept: a known point satisfies the equation, and subtracting mx from its output finds b. The result must reproduce the visible points, regardless of the size of the graph window.

RuleRule: For a nonvertical line, m = , then y = mx + b. Read b at x = 0, or find b = − m from a known point.

The same idea, five ways

Say it

Read two places on the line. Find the change per step, then the output at zero.

Write it

The equation describes the same straight line as the plotted points.

In math

m = $y sub 2 − y sub 1 over x sub 2 − x sub 1$
b = $y sub 1$ − m $x sub 1$
y = mx + b
y − $y sub 1$ = m(x − $x sub 1$ )

A road's tilt and one known landmark let you describe where the road goes.

See it

The crossing gives b = 1, and one step right gives one step up.

The same idea, other ways

As a picture

The line goes through (0, 1) and (1, 2). Its rise is 1 while its run is 1, so m = 1 and b = 1. The equation is y = x + 1.

An intercept and a rise over run describe the whole original line.

As a detective story

If the crossing is hidden, a visible point is still a clue. With slope 2 and point (5, 12), the equation 12 = 2(5) + b leaves b = 2 after subtracting 10. Plug back: 2(5) + 2 = 12. You have found the hidden crossing.

Removing 10 from both sides finds the missing starting output 2.

You are given	Do this
Slope and y-intercept	Write y = mx + b.
Slope and one point	Write y − $y sub 1$ = m(x − $x sub 1$ ), then simplify if needed.
Two points	Find the slope first, then use either point in point-slope form.
A table	Check output change divided by input change on every adjacent interval. Read b at input zero, or solve for it from another column.
A word problem	The starting amount is b. The amount per input unit is m. If those are hidden, use two stated input-output pairs.

.1Slope and intercept supplied

If the tilt and the crossing are already given, you have both ingredients. The slope says how much to add for each input step. The intercept says what you have before any steps. Like a parking meter with a starting charge and a charge per hour, the equation combines those two amounts.

Rule: A known slope m and y-intercept b give y = mx + b, because mx measures the accumulated change and b supplies the starting output.
The intercept point is (0, b), because m × 0 contributes zero.
A negative b starts below zero. It does not make the line decreasing, because the sign of m controls the change.

Slope 3: the coefficient of x

Intercept 5: the separate constant

y = 3x + 5

Keep the slope and the intercept in their different roles.

Reminder

Multiplication by zero. Any number times zero is zero: 3 × 0 = 0, so 3(0) + 5 = 5.

The same idea, five ways

Say it

Start at b, then change by m for each input step.

Write it

The slope and y-intercept determine the line.

In math

y = mx + b
f(0) = b
(0, b)

Start with a parking charge, then add the charge per hour.

See it

The original rule multiplies by the slope before adding the starting output.

Worked exampleInsert two supplied ingredients

Writing a line means combining its starting output and change per input unit. Write the original line with slope 3 and y-intercept 5. Find its outputs at zero and one.

Start at 5 and rise 3 for each step right.

Put m = 3 and b = 5 into y = mx + b to get y = 3x + 5.Both ingredients are supplied, so no missing value needs to be solved.
At x = 0, y = 3(0) + 5 = 5. At x = 1, y = 3(1) + 5 = 8.Zero tests the starting output, and one step tests the change.

Answer

y = 3x + 5
At x = 0: y = 5.
At x = 1: y = 8.

Check The output change is 8 − 5 = 3 for an input change of 1, so the stated slope is recovered.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: Slope 3 and intercept 5 give y = 5x + 3.

This swaps the amount per step with the starting amount. At input zero it gives 3 instead of 5.

✓ Instead: Write y = 3x + 5 and check that x = 0 gives y = 5.

Tips and tricks

Tip: Put the amount per step beside x. Put the starting output by itself.

.2Graph with a visible intercept

A visible crossing at the vertical axis gives the starting output directly. Choose that crossing and another marked point. Read their coordinates from the axis numbers. Then measure the output change and the input change. You are using two landmarks on a straight road, not estimating its tilt from how it looks on the page.

Rule: A visible point (0, b) gives the y-intercept b, because its input is zero.
Two exact points determine the slope, because their output change divided by input change is constant along a line.
A compressed or stretched picture can change the appearance of steepness, so coordinate numbers are more reliable than the drawn angle.

Read b on the vertical axis, where the input is zero.

Reminder

Subtracting a negative. Subtracting a negative adds its positive amount: 1 − (−3) = 1 + 3 = 4.

The same idea, five ways

Say it

Read the crossing, then measure rise divided by run.

Write it

The output at the vertical axis is the starting value.

In math

(0, b)
m = $\frac{r i s e}{r u n}$
y = mx + b

Read the road's starting landmark before measuring its change between landmarks.

See it

The original line crosses at −3 and rises 4 over a run of 2.

Worked exampleRead a crossing below zero

Writing an equation means recovering the pictured line's starting output and change per step. Use the original graph's labeled points (0, −3) and (2, 1).

A negative starting output can belong to an increasing line.

Read b = −3 from the point (0, −3).That point is on the vertical axis where the input is zero.
m = $1 − (−3) over 2 − 0$ = $4 over 2$ = 2.Subtracting −3 adds 3, and the run is two coordinate units.
Write y = 2x − 3.The line starts at −3 and adds 2 per input unit.

Answer

y = 2x − 3

Check At x = 0, y = −3. At x = 2, y = 4 − 3 = 1. The graph's two labels both fit.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: The line crosses the horizontal axis near 1.5, so b = 1.5.

The horizontal crossing has output zero. The y-intercept has input zero. These are different crossings.

✓ Instead: Use the vertical crossing (0, −3), so b = −3.

Tips and tricks

Tip: For b, follow the axis where x = 0 and read the output.

.3Graph with an intercept outside the window

A map can show only part of a road. The road still has a location farther back, even when that place is off the map. A graph works the same way. If the vertical axis is outside the window, use two visible points for the slope. Then use one point to remove the accumulated change and uncover the starting output.

Rule: From a visible point, b = $y sub 1$ − m $x sub 1$ , because subtracting the accumulated change m $x sub 1$ leaves the starting output.
The intercept may be outside the window, because the window shows only selected input and output ranges.
After solving for b, plug it into the point's equation, because the recovered starting value must reproduce the known output.

A graph window edge need not be x = 0

8 = −6 + b

b = 14

−6 + 14 = 8

Recover the hidden intercept with a known point and check it in the same equation.

Reminder

Keeping an equation balanced. Add the same amount to both sides: 8 = −6 + b becomes 14 = b after adding 6.

The same idea, five ways

Say it

Take away the change since zero to recover the hidden start.

Write it

A visible point and the slope determine the unseen intercept.

In math

$y sub 1$ = m $x sub 1$ + b
b = $y sub 1$ − m $x sub 1$
y − $y sub 1$ = m(x − $x sub 1$ )

Work backward from a road landmark to a starting place outside the map.

See it

This original window shows two points but neither x = 0 nor the intercept 14.

Worked exampleRecover a hidden crossing

Finding the equation means describing this line even beyond the displayed window. Use the original graph's points (4, 8) and (8, 2), whose intercept is not shown.

Use the point's coordinates to find the intercept outside this original window.

m = $2 − 8 over 8 − 4$ = $−6 over 4$ = − $3 over 2$ .The output falls by 6 over a run of 4.
Use (4, 8): 8 = − $3 over 2$ (4) + b = −6 + b.The known point lets us find the starting output b without seeing it.
Add 6 to both sides: b = 14. Plug back: −6 + 14 = 8.Adding 6 removes the known contribution and the substitution verifies the recovered intercept.
Write y = − $3 over 2$ x + 14.Both the slope and the starting output are now known.

Answer

m = − $3 over 2$
b = 14
y = − $3 over 2$ x + 14

Check At x = 8, −(8) + 14 = −12 + 14 = 2. The other visible point fits too.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: The left edge of this graph is at output 9.5, so b = 9.5.

The left edge is x = 3, not x = 0. An edge of a graph window is not automatically an axis.

✓ Instead: At x = 3, −1.5(3) + 14 = 9.5. The actual intercept is b = 14 at x = 0.

Tips and tricks

Tip: Write the x-coordinate of the visible left edge before deciding whether it is the y-axis.

Strategy: step by step

1. Read the numbers on both axes, because a drawn square need not mean one coordinate unit.
2. Choose two exact labeled points with different x-coordinates, because a nonzero run is needed for the slope.
3. Calculate output change divided by input change in the same point order, because reversing only one subtraction changes the sign.
4. Read the output where x = 0 if that crossing is visible. Otherwise substitute a known point into y = mx + b and solve for b to find the unseen starting output.
5. Write the equation and substitute both visible points, because a correct equation must give both of their outputs.

Strategy

Strategy: Choose the equation from the information given

Are the slope and y-intercept both given?

YesInsert them into y = mx + b, then check the output at zero.

NoLook for a known point or two readable points.

↓

Is the slope known and one point given?

YesUse y − = m(x − ), or substitute the point and solve for b. Plug the point back in.

NoIf two points are given or visible, calculate the slope first.

↓

Is this a graph with a visible crossing at x = 0?

YesRead b from that crossing and use the slope found from two points.

NoFor a graph with a hidden crossing, use a visible point to calculate b. For a table or story, check the rates and identify the starting amount.

1. Identify whether you have a slope, an intercept, a point, two points, a graph, a table or a word description.
2. Find any missing slope from two points or from a stated amount per input unit.
3. Use y = mx + b when b is known. Use y − $y sub 1$ = m(x − $x sub 1$ ) or solve b = $y sub 1$ − m $x sub 1$ when another point is known.
4. Check the result against the original information, including units and allowed inputs.

Worked exampleRead a small original graph

Writing the equation means finding a rule that gives every output on the pictured line. Find the equation of this original line through (0, 1) and (1, 2).

This original graph has a visible starting output and a one-unit run.

At the vertical axis, x = 0 and y = 1, so b = 1.The y-intercept is the output when the input is zero.
m = $2 − 1 over 1 − 0$ = $1 over 1$ = 1.The output goes up 1 while the input goes up 1.
Write y = 1x + 1 = x + 1.Slope-intercept form uses the slope 1 and the starting output 1.

Answer

y = x + 1

Check At x = 0, 0 + 1 = 1. At x = 1, 1 + 1 = 2. Both plotted points are reproduced.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Rung 1: Read a visible intercept

Finding the equation means giving a rule for the outputs on this original line. Use the marked points (0, 1) and (1, 2).

Start with a visible crossing and a run of one.

Read b = 1 at x = 0.The vertical crossing gives the starting output.
m = $2 − 1 over 1 − 0$ = 1.Rise and run are both 1.
Write y = x + 1.Insert m = 1 and b = 1 into slope-intercept form.

Answer

y = x + 1

Check The formula gives 1 at x = 0 and 2 at x = 1, matching both labels.

Rung 2Rung 2: Count a rise and a longer run

Finding the equation means reading a change per single input unit. Use the original graph's points (0, −2) and (3, 0).

The original line rises 2 coordinate units while moving right 3.

Read b = −2 from (0, −2).The input at this crossing is zero.
m = $0 − (−2) over 3 − 0$ = $2 over 3$ .A rise of 2 is spread over a run of 3, not a run of 1.
Write y = $2 over 3$ x − 2.The fractional slope is the amount added per input unit.

Answer

y = x − 2

Check At x = 3, (3) − 2 = 2 − 2 = 0. At x = 0, the output is −2.

Rung 3Rung 3: A falling line with unequal coordinate changes

Reading the equation means using numbered coordinate values rather than counting drawn squares. The neighboring marked points in this original graph differ by 2 horizontally and 3 vertically. Use the labeled points (2, 6) and (6, 0) to find the equation.

Between neighboring labeled points, a horizontal coordinate step of 2 corresponds to a vertical fall of 3.

Read the coordinate changes: 0 − 6 = −6 vertically and 6 − 2 = 4 horizontally.Across the two intervals between labeled points, the horizontal changes total 2 + 2 = 4 and the vertical falls total 3 + 3 = 6.
m = $−6 over 4$ = − $3 over 2$ .Coordinate rise divided by coordinate run gives slope. This calculation uses labeled numbers and works even when the axes are drawn at different scales.
Use (2, 6): 6 = − $3 over 2$ (2) + b = −3 + b. Add 3 to find b = 9.Removing the known contribution finds the output at input zero.
Plug back: −3 + 9 = 6. Write y = − $3 over 2$ x + 9.The recovered intercept must reproduce the point before we use it.

Answer

y = −x + 9

Check At x = 6, −(6) + 9 = −9 + 9 = 0. A rightward input change of 4 gives an output change of −6, as shown.

Rung 4Rung 4: Recover an unseen intercept

Finding the equation means extending the rule beyond the graph window. The original window shows (4, 8) and (8, 2), but it does not show x = 0. Find the equation and the hidden intercept.

Neither the left edge nor the top edge is the y-intercept of this original line.

m = $2 − 8 over 8 − 4$ = $−6 over 4$ = − $3 over 2$ .The labeled points show a fall of 6 over a run of 4.
Substitute (4, 8): 8 = − $3 over 2$ (4) + b = −6 + b.A visible point lets us solve for the hidden starting output.
Add 6 to get b = 14. Substitute back: −6 + 14 = 8.The equal change on both sides isolates b and the substitution confirms it.
Write y = − $3 over 2$ x + 14 and identify the intercept point (0, 14).At input zero the slope term vanishes, even though that input lies outside this window.

Answer

y = − $3 over 2$ x + 14
The unseen y-intercept is (0, 14).

Check At x = 8, −12 + 14 = 2, matching the other point. At x = 0, the formula gives 14, above the displayed output range.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: Two drawn squares right and two squares down always mean slope −1.

The axes may assign different numbers to each square. Coordinate changes, rather than the picture's square counts, determine slope.

✓ Instead: If the run is 4 units and the fall is 6 units, m = = −.

Tips and tricks

Tip: Write the coordinates of two exact points before touching the slope formula.
Tip: Find m, find b, write the line. Check both points.

Trap. Treating the left edge of the displayed window as the y-axis. The y-axis is where x = 0, which may not be shown.