Part 1 — The screenshot idea, expanded carefully
The screenshot says: stationary points on f become x-intercepts on f′; increasing parts of f become positive parts of f′; decreasing parts of f become negative parts of f′; inflection points on f become turning points on f′.
This is correct, but it is too short. The full meaning is below.
f(x)
f(x) is the original graph.
The y-value of f means the actual height of the graph.
Think: “Where is the road?”
f′(x)
f′(x) is the slope graph.
The y-value of f′ means the slope of f at the same x-value.
Think: “Is the road uphill, downhill, or flat?”
f′′(x)
f′′(x) is the slope-change graph.
The y-value of f′′ tells whether the slope numbers are getting bigger or smaller.
Think: “Is the road bending upward or downward?”
The most important warning
All three graphs use the same x-value, but their y-values mean different things. A point such as x = 2 on f, f′, and f′′ is the same horizontal location, but the vertical meaning changes.
| Graph | x-value means | y-value means | Simple question to ask |
|---|---|---|---|
| f | the same input/location | height | How high is the original graph? |
| f′ | the same input/location | slope of f | Is f going up, going down, or flat? |
| f′′ | the same input/location | change of slope | Are the slope numbers getting bigger or smaller? |
Part 2 — Direct correspondence: f → f′
Stationary point on f
The graph of f is flat.
The graph of f is flat.
→
x-intercept on f′
Because slope = 0.
Because slope = 0.
f is increasing
The graph of f moves upward as x moves right.
The graph of f moves upward as x moves right.
→
f′ is positive
Draw f′ above its x-axis.
Draw f′ above its x-axis.
f is decreasing
The graph of f moves downward as x moves right.
The graph of f moves downward as x moves right.
→
f′ is negative
Draw f′ below its x-axis.
Draw f′ below its x-axis.
Inflection point on f
The graph of f changes bending direction.
The graph of f changes bending direction.
→
Turning point on f′
Because f′ changes from increasing to decreasing, or decreasing to increasing.
Because f′ changes from increasing to decreasing, or decreasing to increasing.
| What you see on f | What it means | What you draw on f′ | Why |
|---|---|---|---|
| f goes up | slope is positive | f′ above x-axis | positive slope |
| f goes down | slope is negative | f′ below x-axis | negative slope |
| f is flat | slope is zero | f′ on x-axis | zero slope |
| f has local maximum | slope changes + to − | f′ crosses from above to below | uphill → flat → downhill |
| f has local minimum | slope changes − to + | f′ crosses from below to above | downhill → flat → uphill |
| f has stationary inflection | slope is 0 but same direction continues | f′ touches x-axis but may not cross | flat for one instant only |
| f has inflection point | concavity changes | f′ has turning point | slope stops decreasing and starts increasing, or opposite |
Part 3 — Direct correspondence: f′ → f and f′′
| What you see on f′ | Meaning for f | Meaning for f′′ |
|---|---|---|
| f′ above x-axis | f is increasing | not enough information by sign alone |
| f′ below x-axis | f is decreasing | not enough information by sign alone |
| f′ = 0 | f has a horizontal tangent | not enough information by zero alone |
| f′ changes + to − | f has a local maximum | depends on local behavior |
| f′ changes − to + | f has a local minimum | depends on local behavior |
| f′ is increasing | f is concave up | f′′ positive |
| f′ is decreasing | f is concave down | f′′ negative |
| f′ has a turning point | f has an inflection point | f′′ = 0 and changes sign |
When the exam gives you the graph of f′, read it like a slope map. Above the axis means f goes up. Below the axis means f goes down. Turning points on f′ become inflection points on f.
Part 4 — Direct correspondence: f′′ → f′ and f
| What you see on f′′ | Meaning for f′ | Meaning for f |
|---|---|---|
| f′′ above x-axis | f′ is increasing | f is concave up |
| f′′ below x-axis | f′ is decreasing | f is concave down |
| f′′ = 0 and changes sign | f′ has a turning point | f has an inflection point |
| f′′ stays positive | f′ keeps increasing | f keeps concave up |
| f′′ stays negative | f′ keeps decreasing | f keeps concave down |
If only f′′ is given, the exact f′ and f are usually not unique. Integrating introduces constants. That means you can often draw one possible f′ and one possible f unless the question gives extra information.
Part 5 — The drawing method with vertical guide lines
1
Stack the graphs vertically. Put f, f′, and f′′ one above another, or use the order given in the question.
2
Mark important x-values first. These include stationary points, x-intercepts, turning points, and inflection points.
3
Draw vertical dashed guide lines. The same x-value must line up across the graphs.
4
Place key points before drawing the smooth curve. Do not randomly draw the curve first.
5
Decide above/below the x-axis. Positive means above. Negative means below.
6
Connect smoothly. Use smooth curves unless the original graph has a corner or discontinuity.
Part 6 — Three dropdown cases with dynamic smooth curves
Case 1 — Given f(x), draw f′(x) and f′′(x)
Theory application
1
Start with f. Ask: where is f flat?
2
Flat points on f become x-intercepts on f′.
3
Where f goes up, f′ is positive. Where f goes down, f′ is negative.
4
Where f has an inflection point, f′ has a turning point.
5
Then read f′ to draw f′′.
Example
f(x)=x³−3x
f′(x)=3x²−3
f′′(x)=6x
| x | On f | On f′ | On f′′ |
|---|---|---|---|
| −1 | local maximum | x-intercept | negative value |
| 0 | inflection point | turning point | x-intercept |
| 1 | local minimum | x-intercept | positive value |
x=-2.5
f
f′
f′′
Explanation
Case 2 — Given f′(x), draw f(x) and f′′(x)
Theory application
1
Start with the given f′ graph. It is a slope graph.
2
If f′ is positive, f goes up. If f′ is negative, f goes down.
3
If f′ = 0, f is flat.
4
If f′ changes positive to negative, f has a local maximum.
5
If f′ changes negative to positive, f has a local minimum.
6
If f′ has a turning point, f has an inflection point.
Example
Given f′(x)=x²−4
One possible f(x)=x³/3−4x
f′′(x)=2x
| x | On f′ | On f | On f′′ |
|---|---|---|---|
| −2 | f′=0, + to − | local maximum | negative |
| 0 | turning point | inflection point | zero |
| 2 | f′=0, − to + | local minimum | positive |
x=-3
given f′
f
f′′
Explanation
Case 3 — Given f′′(x), draw f′(x) and f(x)
Theory application
1
Start with f′′. It tells whether f′ is increasing or decreasing.
2
Where f′′ is positive, draw f′ increasing and f concave up.
3
Where f′′ is negative, draw f′ decreasing and f concave down.
4
If f′′ changes sign, f′ has a turning point and f has an inflection point.
5
Use constants to choose one possible f′ and f.
Example
Given f′′(x)=6x
Choose f′(x)=3x²−3
Then f(x)=x³−3x
| x | On f′′ | On f′ | On f |
|---|---|---|---|
| 0 | zero, sign changes | turning point | inflection point |
| −1 and 1 | not from f′′ directly | f′=0 | stationary points |
x=-2.5
given f′′
chosen f′
chosen f
Explanation
Part 7 — Final memory sentence
f is the road. f′ is the slope of the road. f′′ tells whether the slope is getting bigger or smaller. Mark the same x-values, draw vertical guide lines, place key points, then connect smoothly.