📓 Math Self-Study Notebook
Linear Functions · Arithmetic Sequences · Scatter Plots | 20 Key Practice Problems
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📈 Linear Functions
SLOPE-INTERCEPT FORM \(y = mx + b\)
• m = slope → "steepness / direction of the line"
• b = y-intercept → "where the line crosses the y-axis (x = 0)"
• Slope formula: RISE over RUN \(= \dfrac{y_2 - y_1}{x_2 - x_1}\)
• Positive slope ↗ | Negative slope ↘ | Zero slope → horizontal
• m = slope → "steepness / direction of the line"
• b = y-intercept → "where the line crosses the y-axis (x = 0)"
• Slope formula: RISE over RUN \(= \dfrac{y_2 - y_1}{x_2 - x_1}\)
• Positive slope ↗ | Negative slope ↘ | Zero slope → horizontal
\(y = 2x + 3\) → slope = 2, y-intercept = 3
When x increases by 1, y increases by 2. The line goes up to the right ↗
When x increases by 1, y increases by 2. The line goes up to the right ↗
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⭐ Basic | Reading slope-intercept form
For the linear function \(y = -3x + 5\), which correctly identifies the slope and y-intercept?
In \(y = mx + b\): m is slope, b is y-intercept.
\(y = \mathbf{-3}x + \mathbf{5}\) → slope = −3, y-intercept = 5
⚠️ Common mistake: dropping the negative sign. The slope is −3, not 3!
\(y = \mathbf{-3}x + \mathbf{5}\) → slope = −3, y-intercept = 5
⚠️ Common mistake: dropping the negative sign. The slope is −3, not 3!
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⭐ Basic | Calculating slope from two points
What is the slope of the line passing through the points \((1,\,2)\) and \((3,\,8)\)?
Slope \(= \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{8 - 2}{3 - 1} = \dfrac{6}{2} = \mathbf{3}\)
Remember: RISE ÷ RUN — always divide the change in y by the change in x!
Remember: RISE ÷ RUN — always divide the change in y by the change in x!
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⭐⭐ Intermediate | Writing the equation of a line
A line has slope \(2\) and passes through the point \((1,\,5)\). What is its equation?
Plug \(m = 2\) and point \((1, 5)\) into \(y = mx + b\):
\(5 = 2(1) + b \Rightarrow b = 3\)
∴ \(y = 2x + 3\)
⚠️ B is wrong: don't use the y-value of the point as the y-intercept directly!
\(5 = 2(1) + b \Rightarrow b = 3\)
∴ \(y = 2x + 3\)
⚠️ B is wrong: don't use the y-value of the point as the y-intercept directly!
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⭐⭐ Intermediate | x-intercept vs y-intercept
What is the x-intercept of the linear function \(y = 4x - 8\)?
Hint: the x-intercept is where the line crosses the x-axis — set y = 0!
Hint: the x-intercept is where the line crosses the x-axis — set y = 0!
x-intercept: set \(y = 0\) → \(0 = 4x - 8 \Rightarrow x = 2\)
y-intercept: set \(x = 0\) → \(y = -8\)
⚠️ Trick alert: x-intercept → y = 0 | y-intercept → x = 0
y-intercept: set \(x = 0\) → \(y = -8\)
⚠️ Trick alert: x-intercept → y = 0 | y-intercept → x = 0
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⭐⭐ Intermediate | Behaviour of graphs
Which statement is true about a linear function with a negative slope?
Negative slope → graph falls from left to right ↘
• POSITIVE slope: rises ↗ — x↑ means y↑
• NEGATIVE slope: falls ↘ — x↑ means y↓
Memory: "Positive = going UP a hill, Negative = going DOWN a hill"
• POSITIVE slope: rises ↗ — x↑ means y↑
• NEGATIVE slope: falls ↘ — x↑ means y↓
Memory: "Positive = going UP a hill, Negative = going DOWN a hill"
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⭐⭐⭐ Advanced | Parallel lines
Which line is parallel to \(y = 3x + 1\) and passes through the point \((0,\,-2)\)?
Parallel lines have the same slope but different y-intercepts!
Parallel lines have the same slope but different y-intercepts!
Parallel → keep slope m = 3. Point (0, −2) is the y-intercept directly!
∴ \(y = 3x + (-2) = 3x - 2\)
⚠️ C (\(y = 3x + 1\)) is the original line — same line, not parallel!
PARALLEL = same slope, DIFFERENT intercept
∴ \(y = 3x + (-2) = 3x - 2\)
⚠️ C (\(y = 3x + 1\)) is the original line — same line, not parallel!
PARALLEL = same slope, DIFFERENT intercept
🔢 Arithmetic Sequences
ARITHMETIC = ADD the same number each time
• d = common difference (added each time)
• nth term: \(a_n = a_1 + (n-1)d\) → "start + how many jumps × jump size"
• Sum: \(S_n = \dfrac{n(a_1 + a_n)}{2}\) → "count × average of first & last"
• Middle term (arithmetic mean): \(b = \dfrac{a + c}{2}\)
• d = common difference (added each time)
• nth term: \(a_n = a_1 + (n-1)d\) → "start + how many jumps × jump size"
• Sum: \(S_n = \dfrac{n(a_1 + a_n)}{2}\) → "count × average of first & last"
• Middle term (arithmetic mean): \(b = \dfrac{a + c}{2}\)
Sequence: 2, 5, 8, 11, … → common difference d = 3
nth term: \(a_n = 2 + (n-1) \times 3 = 3n - 1\)
5th term: \(a_5 = 3(5) - 1 = 14\) ✓
nth term: \(a_n = 2 + (n-1) \times 3 = 3n - 1\)
5th term: \(a_5 = 3(5) - 1 = 14\) ✓
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⭐ Basic | Finding the common difference
What is the common difference of the sequence \(3,\; 7,\; 11,\; 15,\; \ldots\)?
Subtract consecutive terms: \(7 - 3 = 4\), \(11 - 7 = 4\), \(15 - 11 = 4\)
Common difference \(d = \mathbf{4}\)
⚠️ The common difference is NOT the first term — it's the gap between terms!
Common difference \(d = \mathbf{4}\)
⚠️ The common difference is NOT the first term — it's the gap between terms!
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⭐⭐ Intermediate | nth term formula
An arithmetic sequence has first term \(5\) and common difference \(-2\). What is the 10th term?
\(a_n = a_1 + (n-1)d\)
\(a_{10} = 5 + (10-1)(-2) = 5 + 9 \times (-2) = 5 - 18 = \mathbf{-13}\)
⚠️ It's \((n-1)\), NOT \(n\)! For the 10th term, you add the difference 9 times!
\(a_{10} = 5 + (10-1)(-2) = 5 + 9 \times (-2) = 5 - 18 = \mathbf{-13}\)
⚠️ It's \((n-1)\), NOT \(n\)! For the 10th term, you add the difference 9 times!
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⭐⭐ Intermediate | Sum of arithmetic sequence
Find the sum of the sequence \(1,\; 3,\; 5,\; 7,\; \ldots,\; 19\).
(All odd numbers from 1 to 19)
(All odd numbers from 1 to 19)
Count: \(1, 3, 5, \ldots, 19\) → common difference 2, so \(n = 10\) terms.
\(S_n = \dfrac{n(a_1 + a_n)}{2} = \dfrac{10(1 + 19)}{2} = \dfrac{200}{2} = \mathbf{100}\)
Memory: SUM = count × (first + last) ÷ 2
\(S_n = \dfrac{n(a_1 + a_n)}{2} = \dfrac{10(1 + 19)}{2} = \dfrac{200}{2} = \mathbf{100}\)
Memory: SUM = count × (first + last) ÷ 2
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⭐⭐ Intermediate | Finding which term a value is
In an arithmetic sequence with first term \(2\) and common difference \(3\), which term equals \(50\)?
\(a_n = 2 + (n-1) \times 3 = 50\)
\((n-1) \times 3 = 48 \Rightarrow n - 1 = 16 \Rightarrow n = \mathbf{17}\)
⚠️ Don't forget to add 1 at the end! If \(n - 1 = 16\), then \(n = 17\), not 16!
\((n-1) \times 3 = 48 \Rightarrow n - 1 = 16 \Rightarrow n = \mathbf{17}\)
⚠️ Don't forget to add 1 at the end! If \(n - 1 = 16\), then \(n = 17\), not 16!
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⭐⭐⭐ Advanced | Arithmetic mean
The three numbers \(x,\; 10,\; 16\) form an arithmetic sequence in that order. Find \(x\).
Hint: the middle term = average of the other two!
Hint: the middle term = average of the other two!
Arithmetic mean: middle term = average of its neighbours
\(10 = \dfrac{x + 16}{2} \Rightarrow 20 = x + 16 \Rightarrow x = \mathbf{4}\)
Check: 4, 10, 16 → difference = 6 each time ✓
\(10 = \dfrac{x + 16}{2} \Rightarrow 20 = x + 16 \Rightarrow x = \mathbf{4}\)
Check: 4, 10, 16 → difference = 6 each time ✓
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⭐⭐ Intermediate | Real-world pattern
Equilateral triangles are built using matchsticks placed end-to-end.
1 triangle needs 3 sticks, 2 triangles need 5 sticks, 3 need 7 sticks, …
How many matchsticks are needed for \(n\) triangles?
1 triangle needs 3 sticks, 2 triangles need 5 sticks, 3 need 7 sticks, …
How many matchsticks are needed for \(n\) triangles?
Pattern: 3, 5, 7, … → first term = 3, common difference = 2
\(a_n = 3 + (n-1) \times 2 = 2n + 1\)
Check: n=2 → \(2(2)+1 = 5\) ✓ | n=3 → \(2(3)+1 = 7\) ✓
\(a_n = 3 + (n-1) \times 2 = 2n + 1\)
Check: n=2 → \(2(2)+1 = 5\) ✓ | n=3 → \(2(3)+1 = 7\) ✓
🔵 Scatter Plots & Correlation
CORRELATION = how two variables move together
• POSITIVE correlation: points trend up-right ↗ (both increase together)
• NEGATIVE correlation: points trend down-right ↘ (one up → other down)
• NO correlation: points scattered randomly — no pattern
• STRONG: points cluster tightly near the trend line
• WEAK: points spread far from the trend line
• POSITIVE correlation: points trend up-right ↗ (both increase together)
• NEGATIVE correlation: points trend down-right ↘ (one up → other down)
• NO correlation: points scattered randomly — no pattern
• STRONG: points cluster tightly near the trend line
• WEAK: points spread far from the trend line
Height (x) vs. Weight (y): taller people tend to weigh more
→ Points trend up and to the right = Positive correlation
→ Points trend up and to the right = Positive correlation
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⭐ Basic | Identifying correlation type
In a scatter plot, points trend upward to the right (\(\nearrow\)). What does this indicate?
Up-right trend ↗ → as x increases, y also increases → Positive correlation
POSITIVE = up-right ↗
NEGATIVE = down-right ↘
POSITIVE = up-right ↗
NEGATIVE = down-right ↘
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⭐⭐ Intermediate | Real-world correlation
Which of the following is an example of a negative correlation?
Negative correlation: one variable up → other variable down.
C: temperature ↑ → hot drinks ↓ → Negative correlation ✓
A & B: both positive. D: physically incorrect (exercise reduces body fat).
C: temperature ↑ → hot drinks ↓ → Negative correlation ✓
A & B: both positive. D: physically incorrect (exercise reduces body fat).
15
⭐⭐ Intermediate | Strength of correlation
In a scatter plot, the closer the points cluster to a straight line, the correlation is…?
CLOSE to line → STRONG correlation
SPREAD far from line → WEAK correlation
Strength describes how reliably one variable predicts the other — it's independent of direction (positive or negative).
SPREAD far from line → WEAK correlation
Strength describes how reliably one variable predicts the other — it's independent of direction (positive or negative).
16
⭐⭐ Intermediate | No correlation
Which best describes a scatter plot that shows no correlation?
No correlation = NO clear pattern or direction in the scatter plot.
The points appear randomly distributed — knowing x tells you nothing about y.
Example: shoe size vs. math score → no relationship → random scatter.
The points appear randomly distributed — knowing x tells you nothing about y.
Example: shoe size vs. math score → no relationship → random scatter.
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⭐⭐⭐ Advanced | Outliers
In a scatter plot, a data point that lies far away from the rest of the points is called a(n)…?
An OUTLIER is a data point that falls far outside the general pattern.
Outliers can distort the trend line and mislead your interpretation of the correlation.
Memory: "OUT" + "LIER" = a point lying outside the group
Outliers can distort the trend line and mislead your interpretation of the correlation.
Memory: "OUT" + "LIER" = a point lying outside the group
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⭐⭐ Intermediate | Trend line
A straight line drawn through a scatter plot that best represents the overall pattern of the data is called a…?
A TREND LINE (also called line of best fit) passes through the middle of the data points, showing the general direction.
• Trend line rising ↗ → positive correlation
• Trend line falling ↘ → negative correlation
Use the trend line to predict values!
• Trend line rising ↗ → positive correlation
• Trend line falling ↘ → negative correlation
Use the trend line to predict values!
19
⭐⭐⭐ Advanced | Interpreting correlation
A survey of students shows that as hours of sleep (x) increase, their focus level in class (y) also tends to increase. Which correctly describes the trend line and correlation?
Sleep ↑ → focus ↑ → both increase together
= Positive correlation = trend line goes up-right ↗
⚠️ D has the correct trend line direction, but the wrong correlation label! Direction ↗ is always positive, not negative.
= Positive correlation = trend line goes up-right ↗
⚠️ D has the correct trend line direction, but the wrong correlation label! Direction ↗ is always positive, not negative.
20
⭐⭐⭐ Advanced | Spot the error
Which of the following statements about scatter plots is INCORRECT?
D is INCORRECT!
Negative correlation means one variable increases while the other decreases.
"Both increase together" = positive correlation.
A ✓ B ✓ C ✓ D ✗
NEGATIVE = OPPOSITE directions
Negative correlation means one variable increases while the other decreases.
"Both increase together" = positive correlation.
A ✓ B ✓ C ✓ D ✗
NEGATIVE = OPPOSITE directions