🎯 Learning Objectives
- How to read a scatterplot, line graph, and bar graph.
- Positive, negative, and no association at a glance.
- How to estimate slope and intercept from a line of best fit: and what they mean in context.
- How to identify outliers and how they affect the model.
- 10 SAT-style practice questions.
1. Reading Graphs Like a Pro
Roughly one in seven SAT Math questions involves a graph. Most of these reduce to four steps: identify the axes, identify the units, find the relevant data point, and answer the question.
STEP 1: Read both axis labels (with units)
STEP 2: Identify what the question is asking
STEP 3: Locate the data point or read the line
STEP 4: Match your reading to an answer choice
2. Three Types of Association
Pattern | What It Looks Like | Meaning |
Positive | Points slope up | As x increases, y increases |
Negative | Points slope down | As x increases, y decreases |
None | Random scatter | x and y are unrelated |
3. Lines of Best Fit
A line of best fit summarizes a scatterplot's overall trend in a single line. The SAT will draw it for you: your job is to read or interpret it, not to compute it.
The line of best fit (slope-intercept form)
y = m · x + b
What the Slope (m) Means
The slope tells you how much y changes per one-unit increase in x. Always read it in context: 'For each additional hour studied, the predicted score increases by 5 points.'
What the Y-Intercept (b) Means
The y-intercept is the predicted value of y when x = 0. In real-world contexts, this is often the 'starting amount': for example, the cost of an item before any units have been added, or the population at year zero.
💡 Pro Tip: Always Use Units
If a question asks what the slope represents, your answer must include units like 'dollars per year' or 'students per classroom'. The right answer choice always makes sense with units.
4. Outliers and How They Affect the Line
An outlier is a point that's far from the rest of the data. A single outlier can pull the line of best fit toward it, changing both the slope and the intercept: sometimes dramatically.
❓ Where is the outlier? | ||
✅ At the END of the data range ▼ | ❌ Near the MIDDLE ▼ | |
It tilts (changes slope) the line strongly | It barely affects slope; mostly nudges intercept | |
5. Predicted vs. Actual
If a real data point sits ABOVE the line of best fit, the model UNDERESTIMATED that point. If it sits BELOW, the model OVERESTIMATED it. The vertical distance from the point to the line is called the residual.
Residual = Actual − Predicted
⚠️ Common Mistakes
'Above the line' means actual is HIGHER than predicted (positive residual).
'Below the line' means actual is LOWER than predicted (negative residual).
Don't confuse this with the visual position of dots above or below other dots.
6. Linear vs. Exponential Growth
On graphs, distinguish:
- Linear: straight line; equal increases over equal intervals
- Exponential: curve that bends sharply upward (or decays sharply); a constant percentage rate of change
- Quadratic: U-shaped or upside-down U; symmetric
7. Strategies
⚡ Read both axes BEFORE looking at the graph
Most graph mistakes come from misreading 'thousands' vs. 'millions' or hours vs. minutes.
⚡ Use rise/run on grid lines
Pick two clean lattice points on the line of best fit and count squares to get slope.
⚡ Eliminate weird answer choices
If the slope is positive, immediately rule out any choice that says 'decreases'.
⚡ Translate slope into a sentence
Saying '$3 per pound' out loud helps you match the right wording in answer choices.
⚡ For 'best model' questions, look at the curve shape
Straight → linear; bending up → exponential growth; bending down → exponential decay.
8. Summary
📌 Key Takeaways
- Read axes first; always note units.
- Slope = change in y per unit change in x.
- Intercept = value when x = 0.
- Residual = actual − predicted (above line = +).
- Outliers near the ends shift slope the most.
- Curve shape: straight = linear, bending = exponential or quadratic.
9. Practice: 10 Questions
Q1. A scatterplot shows hours studied vs. test score with a positive trend. The slope of the best-fit line is 4. What does this slope mean?
A) Score 4 to start B) Score increases by 4 per extra hour C) 4 students per hour D) None
Hint: Slope = change in y per unit x.
Q2. A best-fit line is y = 2x + 5. What does the y-intercept represent?
A) Score after 5 hours B) Score with 0 hours C) Slope D) Number of students
Hint: Intercept = y when x = 0.
Q3. A scatterplot shows no association. Which best describes it?
A) Strong positive B) Strong negative C) Random scatter D) Linear
Hint: No pattern = no association.
Q4. For y = 3x + 10, the actual value at x = 2 is 20. What is the residual?
A) −16 B) −4 C) 4 D) 16
Hint: Predicted = 16; residual = 20 − 16 = 4.
Q5. A point sits BELOW the best-fit line. The model has…
A) Underestimated it B) Overestimated it C) Predicted exactly D) Cannot tell
Hint: Below means actual < predicted → overestimated.
Q6. The slope of a line of best fit is −2. As x increases by 1, y…
A) increases by 2 B) decreases by 2 C) doubles D) stays same
Hint: Negative slope → y decreases.
Q7. Which graph type best models population growth at 5% per year?
A) Linear B) Exponential C) Quadratic D) Constant
Hint: Constant percent → exponential.
Q8. The best-fit line for cost vs. weight is y = 1.5x + 2. What does $1.50 represent?
A) Fixed fee B) Cost per unit weight C) Discount D) Tax rate
Hint: Slope = cost per pound.
Q9. A scatterplot has 10 points clustered along a line plus 1 point far away. Removing the far point will…
A) Have no effect B) Make the fit better C) Worsen the fit D) Change the units
Hint: Removing an outlier usually tightens the fit.
Q10. A line of best fit predicts y = 50 at x = 10. The actual value is 47. The residual is…
A) +3 B) −3 C) +47 D) +50
Hint: 47 − 50 = −3.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) Score increases by 4 per extra hour | Slope = Δy / Δx. |
Q2. | B) Score with 0 hours | Intercept = y when x=0. |
Q3. | C) Random scatter | No association = no pattern. |
Q4. | C) 4 | Residual = actual − predicted. |
Q5. | B) Overestimated | Below line = model too high. |
Q6. | B) decreases by 2 | Negative slope. |
Q7. | B) Exponential | Constant percent → exponential. |
Q8. | B) Cost per unit weight | Slope is rate per unit x. |
Q9. | B) Make the fit better | Removing outlier reduces error. |
Q10. | B) −3 | 47 − 50. |