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Grade 9 Prep Guide

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Adaptive Math Test — Grade 9 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 9 adaptive math assessment. None of the practice problems below are taken from the test bank — they are fresh examples designed to teach the same skills.

For each topic you'll find:

  • A short explanation of the key idea
  • Multiple worked examples (problems solved step by step)
  • Common mistakes to watch out for
  • Practice problems with answers at the end of each section

Table of Contents

Algebra

  1. Solving Linear Equations
  2. Literal Equations
  3. Compound and Absolute Value Inequalities
  4. Systems of Equations

Quadratics 5. Factoring Trinomials 6. Factoring Special Cases 7. Solving Quadratics by Factoring 8. The Quadratic Formula and the Discriminant 9. Vertex Form & Completing the Square 10. Graphing Quadratics

Functions 11. Evaluating Functions 12. Domain and Range 13. Composition of Functions 14. Inverse Functions

Geometry 15. Triangle Congruence 16. Special Right Triangles 17. Circle Theorems 18. Coordinate Geometry

Statistics 19. Sampling Methods and Bias 20. Normal Distribution 21. Expected Value 22. Linear Regression

1. Solving Linear Equations

Key Concepts

Move all variable terms to one side, all constants to the other, then divide by the coefficient. Watch for distributing across parentheses and clearing fractions.

Worked Examples

Example 1. 4(x − 3) + 2x = 8. Solution. 4x − 12 + 2x = 8 → 6x = 20 → x = 10/3.

Example 2. 2x/3 + 1 = 5. Solution. 2x/3 = 4 → x = 6.

Example 3. 3(x − 2) = 5x − 4. Solution. 3x − 6 = 5x − 4 → −2 = 2x → x = −1.

Example 4. A linear equation with no solution? 2x + 5 = 2x − 3 → 5 = −3 (false) → no solution.

Common Mistakes

  • Forgetting to distribute across all terms.
  • Sign errors when moving terms across the = sign.

Practice

  1. 5(x + 2) = 25
  2. 3x + 7 = x − 11
  3. (x − 1)/2 + (x + 3)/3 = 4
  4. 4(x − 2) − x = 2x − 8

Answers: 1. x = 3 | 2. x = −9 | 3. x = 3 | 4. x = 0

2. Literal Equations

Key Concepts

A literal equation has multiple variables. Solving means isolating the requested variable.

Worked Examples

Example 1. Solve A = lw for w. Solution. w = A/l.

Example 2. Solve y = mx + b for x. Solution. x = (y − b)/m.

Example 3. Solve V = (1/3)πr²h for h. Solution. h = 3V/(πr²).

Example 4. Solve C = (5/9)(F − 32) for F. Solution. F = (9/5)C + 32.

Common Mistakes

  • Trying to solve for a variable that appears on both sides without combining first.
  • Skipping distribution of the inverse operation.

Practice

  1. Solve P = 2(l + w) for l.
  2. Solve d = rt for t.
  3. Solve x² + y² = r² for y (positive root).

Answers: 1. l = P/2 − w | 2. t = d/r | 3. y = √(r² − x²)

3. Compound and Absolute Value Inequalities

Key Concepts

  • A compound inequality joins two with "and" (intersection) or "or" (union).
  • For |x| < a (a > 0): −a < x < a.
  • For |x| > a: x < −a or x > a.

Worked Examples

Example 1. Solve −3 < 2x + 1 < 7. Solution. Subtract 1: −4 < 2x < 6. Divide by 2: −2 < x < 3.

Example 2. Solve |x − 4| < 3. Solution. −3 < x − 4 < 3 → 1 < x < 7.

Example 3. Solve |2x + 1| ≥ 9. Solution. 2x + 1 ≤ −9 or 2x + 1 ≥ 9 → x ≤ −5 or x ≥ 4.

Example 4. Solve x < 2 or x > 5. (Already simplified — describes union.)

Common Mistakes

  • Splitting absolute value inequalities incorrectly (< vs > flips the connective from "and" to "or").
  • Forgetting that |x| < 0 has no solution.

Practice

  1. Solve −1 ≤ 3x + 5 < 14.
  2. Solve |x + 2| ≤ 6.
  3. Solve |3x − 1| > 8.
  4. Solve |x| < −1.

Answers: 1. −2 ≤ x < 3 | 2. −8 ≤ x ≤ 4 | 3. x < −7/3 or x > 3 | 4. No solution

4. Systems of Equations

Key Concepts

Three solution types:

  • Unique solution (lines cross).
  • No solution (parallel lines).
  • Infinitely many (same line).

Methods: graphing, substitution, elimination, matrices (basic).

Worked Examples

Example 1. Solve y = 2x − 1 and 3x + y = 14. Solution. Sub: 3x + 2x − 1 = 14 → 5x = 15 → x = 3, y = 5.

Example 2. Solve 2x + 3y = 12 and 4x − 3y = 6 by elimination. Solution. Add: 6x = 18 → x = 3, y = 2.

Example 3. Solve x + 2y = 4 and 2x + 4y = 5. Solution. Multiply 1st by 2: 2x + 4y = 8. Contradicts 2x + 4y = 5 → no solution.

Example 4. Solve 3x − y = 2 and 6x − 2y = 4. Solution. 2nd is 2 × 1st → same line → infinitely many solutions.

Common Mistakes

  • Substituting back into only one equation (always verify both).
  • Forgetting to multiply both sides of an equation when scaling for elimination.

Practice

  1. y = x + 4 and 2x + y = 10.
  2. 3x + 2y = 11 and 5x − 2y = 5.
  3. x − y = 3 and 2x − 2y = 6.

Answers: 1. (2, 6) | 2. (2, 2.5) | 3. Infinitely many

5. Factoring Trinomials

Key Concepts

For ax² + bx + c when a = 1: find two numbers that multiply to c and add to b.

When a > 1: try the AC method (multiply a × c, find two numbers that multiply to that and add to b, then split bx and factor by grouping).

Worked Examples

Example 1. Factor x² + 7x + 10. Solution. 5 × 2 = 10, 5 + 2 = 7 → (x + 5)(x + 2).

Example 2. Factor x² − 5x + 6. Solution. −2 × −3 = 6, −2 + −3 = −5 → (x − 2)(x − 3).

Example 3. Factor x² + x − 12. Solution. 4 × −3 = −12, 4 + (−3) = 1 → (x + 4)(x − 3).

Example 4. Factor 2x² + 7x + 3 (AC = 6 → 1, 6). Solution. 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).

Common Mistakes

  • Sign errors in identifying the two factors.
  • Forgetting to factor out a GCF first when one exists.

Practice

  1. x² + 9x + 14
  2. x² − 11x + 24
  3. x² − 4x − 21
  4. 3x² + 10x + 8

Answers: 1. (x + 7)(x + 2) | 2. (x − 8)(x − 3) | 3. (x − 7)(x + 3) | 4. (3x + 4)(x + 2)

6. Factoring Special Cases

Key Concepts

  • Difference of squares: a² − b² = (a + b)(a − b).
  • Perfect square trinomial: a² + 2ab + b² = (a + b)², a² − 2ab + b² = (a − b)².
  • Sum of squares (a² + b²) does NOT factor over the reals.

Worked Examples

Example 1. x² − 25 = (x + 5)(x − 5).

Example 2. 9x² − 16 = (3x + 4)(3x − 4).

Example 3. x² + 6x + 9 = (x + 3)².

Example 4. 25x² − 30x + 9 = (5x − 3)².

Common Mistakes

  • Trying to factor a sum of squares as a product (it can't be done with reals).
  • Missing the perfect-square pattern.

Practice

  1. x² − 36
  2. 49 − 4x²
  3. x² + 14x + 49
  4. 16x² − 40x + 25

Answers: 1. (x + 6)(x − 6) | 2. (7 + 2x)(7 − 2x) | 3. (x + 7)² | 4. (4x − 5)²

7. Solving Quadratics by Factoring

Key Concepts

If a product equals zero, at least one factor is zero. Set each factor equal to zero and solve.

Worked Examples

Example 1. Solve x² + 5x + 6 = 0. Solution. (x + 2)(x + 3) = 0 → x = −2 or −3.

Example 2. Solve x² − 4x = 0. Solution. x(x − 4) = 0 → x = 0 or 4.

Example 3. Solve x² = 9. Solution. x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = ±3.

Example 4. Solve 2x² + x − 6 = 0. Solution. (2x − 3)(x + 2) = 0 → x = 3/2 or −2.

Common Mistakes

  • Forgetting that a quadratic typically has two solutions.
  • Setting the entire expression equal to one variable instead of zero.

Practice

  1. x² − 7x + 10 = 0
  2. x² − 16 = 0
  3. x² + 6x = −5
  4. 3x² − 12 = 0

Answers: 1. 2, 5 | 2. ±4 | 3. −1, −5 | 4. ±2

8. The Quadratic Formula and the Discriminant

Key Concepts

For ax² + bx + c = 0 (a ≠ 0): x = [−b ± √(b² − 4ac)] / (2a).

The discriminant is b² − 4ac:

  • 0 → two real solutions.

  • = 0 → one real (double) solution.
  • < 0 → two complex solutions (no real).

Worked Examples

Example 1. Solve 2x² + 5x − 3 = 0. Solution. a = 2, b = 5, c = −3. Discriminant = 25 + 24 = 49. x = (−5 ± 7)/4 = 1/2 or −3.

Example 2. Solve x² − 4x + 1 = 0. Solution. Disc = 16 − 4 = 12. x = (4 ± 2√3)/2 = 2 ± √3.

Example 3. Discriminant of x² + 2x + 5? −16 → no real solutions (complex roots).

Example 4. Discriminant of x² − 6x + 9? 0 → one repeated root (x = 3).

Common Mistakes

  • Sign errors with b in −b.
  • Not simplifying the radical at the end.

Practice

  1. Solve x² + 3x − 10 = 0.
  2. Solve 4x² − 12x + 9 = 0.
  3. Discriminant of 2x² + 3x + 5 — how many real roots?
  4. Solve x² + 6x + 4 = 0.

Answers: 1. 2, −5 | 2. 3/2 (double) | 3. Negative → 0 real roots | 4. −3 ± √5

9. Vertex Form & Completing the Square

Key Concepts

Vertex form: y = a(x − h)² + k. Vertex is at (h, k).

Completing the square: for x² + bx, add (b/2)² to make it (x + b/2)². Used to convert standard form to vertex form.

Worked Examples

Example 1. Convert y = x² − 6x + 11 to vertex form. Solution. x² − 6x = (x − 3)² − 9. y = (x − 3)² − 9 + 11 = (x − 3)² + 2. Vertex (3, 2).

Example 2. Solve x² + 8x − 9 = 0 by completing the square. Solution. x² + 8x = 9 → (x + 4)² = 25 → x + 4 = ±5 → x = 1 or −9.

Example 3. Vertex of y = 2(x + 3)² − 4? Solution. (−3, −4).

Example 4. Convert y = x² + 4x + 1 to vertex form. Solution. (x + 2)² − 3.

Common Mistakes

  • Adding (b/2)² but forgetting to subtract it elsewhere to keep the equation balanced.
  • Reading vertex as (h, k) but writing (−h, k).

Practice

  1. Vertex of y = (x − 5)² + 2.
  2. Convert y = x² + 10x + 15.
  3. Solve x² − 6x − 7 = 0 by completing the square.

Answers: 1. (5, 2) | 2. (x + 5)² − 10 | 3. x = 7 or −1

10. Graphing Quadratics

Key Concepts

A parabola y = ax² + bx + c:

  • Opens up if a > 0; down if a < 0.
  • Vertex at x = −b/(2a).
  • Axis of symmetry: vertical line through the vertex.
  • y-intercept: c (when x = 0).
  • x-intercepts: solutions of ax² + bx + c = 0.

Worked Examples

Example 1. Graph y = x² − 4x + 3. Vertex? Solution. x = 4/2 = 2; y = 4 − 8 + 3 = −1. Vertex (2, −1). Roots at x = 1 and x = 3.

Example 2. y = −2x² + 4. Direction & y-intercept? Solution. Opens down; y-intercept 4.

Example 3. Axis of symmetry of y = x² − 6x + 5. Solution. x = 3.

Example 4. Minimum value of y = x² − 8x + 18? Solution. Vertex x = 4; y = 16 − 32 + 18 = 2 (min, opens up).

Common Mistakes

  • Forgetting the sign of −b/(2a).
  • Confusing vertex (single point) with axis (line).

Practice

  1. Vertex of y = x² + 6x + 8.
  2. Direction y = −3x² + 1?
  3. y-intercept of y = 2x² − 5x + 7.
  4. Roots of y = x² − 5x + 6.

Answers: 1. (−3, −1) | 2. Down | 3. 7 | 4. 2 and 3

11. Evaluating Functions

Key Concepts

f(a) means substitute a for every x in the function. Watch parentheses, especially for negative inputs and squared terms.

Worked Examples

Example 1. f(x) = x² − 3x + 1. Find f(2). Solution. 4 − 6 + 1 = −1.

Example 2. g(x) = √(x + 5). Find g(4). Solution. √9 = 3.

Example 3. f(x) = 3x² + 2. Find f(−3). Solution. 3(9) + 2 = 29.

Example 4. h(x) = 5/x. Find h(0). Solution. Undefined.

Common Mistakes

  • Forgetting that (−3)² = 9, not −9.
  • Skipping the order of operations.

Practice

  1. f(x) = x³ − x. Find f(−2).
  2. g(x) = 2x² + 3x − 1. Find g(4).
  3. f(x) = (x + 1)/(x − 1). Find f(3).
  4. f(x) = √(2x − 4). Find f(10).

Answers: 1. −6 | 2. 43 | 3. 2 | 4. 4

12. Domain and Range

Key Concepts

  • Domain: all valid inputs. Watch for division by 0, even roots of negatives, log of non-positive.
  • Range: all possible outputs. Often inferred from graph shape (parabola has a min/max → bounded range).

Worked Examples

Example 1. Domain of f(x) = 1/(x − 3). Solution. All real except x ≠ 3.

Example 2. Domain of f(x) = √(x + 2). Solution. x + 2 ≥ 0 → x ≥ −2.

Example 3. Range of f(x) = x² + 5? Solution. Minimum at x = 0 → y ≥ 5.

Example 4. Range of f(x) = −|x| + 3? Solution. Max at x = 0 → y ≤ 3.

Common Mistakes

  • Forgetting domain restrictions stack (a function can have multiple).
  • Confusing domain (x) and range (y).

Practice

  1. Domain of f(x) = 1/(x² − 4).
  2. Domain of f(x) = √(5 − x).
  3. Range of f(x) = (x − 1)² + 2.
  4. Range of f(x) = −x² + 6.

Answers: 1. x ≠ ±2 | 2. x ≤ 5 | 3. y ≥ 2 | 4. y ≤ 6

13. Composition of Functions

Key Concepts

(f ∘ g)(x) = f(g(x)). Plug g(x) into f wherever you see x.

Order matters: f(g(x)) is usually different from g(f(x)).

Worked Examples

Example 1. f(x) = 2x + 1, g(x) = x². Find f(g(3)). Solution. g(3) = 9; f(9) = 19.

Example 2. Same f, g. Find g(f(3)). Solution. f(3) = 7; g(7) = 49.

Example 3. f(x) = √x, g(x) = x + 5. Find (f ∘ g)(11). Solution. g(11) = 16; f(16) = 4.

Example 4. f(x) = 3x − 2, g(x) = (x + 2)/3. Find f(g(x)). Solution. 3 × (x + 2)/3 − 2 = (x + 2) − 2 = x. (g is the inverse of f.)

Common Mistakes

  • Plugging into the wrong function.
  • Adding functions instead of composing.

Practice

  1. f(x) = x − 4, g(x) = 2x. Find f(g(3)).
  2. Same. Find g(f(3)).
  3. f(x) = x², g(x) = x + 1. Find (f ∘ g)(2).

Answers: 1. 2 | 2. −2 | 3. 9

14. Inverse Functions

Key Concepts

The inverse f⁻¹(x) reverses what f does: f(f⁻¹(x)) = x.

To find: swap x and y, then solve for y.

A function has an inverse only if it's one-to-one (passes the horizontal line test).

Worked Examples

Example 1. Inverse of y = 2x + 3. Solution. x = 2y + 3 → y = (x − 3)/2. So f⁻¹(x) = (x − 3)/2.

Example 2. Inverse of y = x³. Solution. y = ³√x.

Example 3. Inverse of y = (x + 5)/4. Solution. x = (y + 5)/4 → y = 4x − 5.

Example 4. Does y = x² have an inverse over all reals? Solution. No — fails horizontal line test (restrict domain to x ≥ 0 to get one).

Common Mistakes

  • Swapping x and y inconsistently.
  • Calling 1/f(x) the inverse (it's the reciprocal, not the inverse function).

Practice

  1. Inverse of f(x) = 5x − 7.
  2. Inverse of f(x) = (x − 1)/3.
  3. Does f(x) = |x| have an inverse over all reals?

Answers: 1. (x + 7)/5 | 2. 3x + 1 | 3. No

15. Triangle Congruence

Key Concepts

Triangles are congruent if they have the same shape and size. Five tests:

  • SSS (all three sides)
  • SAS (two sides + included angle)
  • ASA (two angles + included side)
  • AAS (two angles + a non-included side)
  • HL (hypotenuse-leg, for right triangles)

Worked Examples

Example 1. Triangles with all three sides 5, 7, 9 each — congruent? Yes (SSS).

Example 2. A triangle with sides 6, 8 and included angle 50°. Another with the same. SAS → congruent.

Example 3. Two right triangles with hypotenuse 10 and one leg 6 — congruent? Yes (HL).

Example 4. SSA — does it always work? No — SSA is NOT a valid congruence test.

Common Mistakes

  • Using AAA as a congruence criterion (it's only similarity).
  • Forgetting that the angle must be included for SAS.

Practice

Identify the test (or say "not enough"):

  1. △ABC and △DEF: ∠A = ∠D, ∠B = ∠E, AB = DE.
  2. AB = DE, BC = EF, CA = FD.
  3. ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
  4. AB = DE, ∠A = ∠D, AC = DF.

Answers: 1. ASA | 2. SSS | 3. Not enough (only similar) | 4. SAS

16. Special Right Triangles

Key Concepts

  • 45-45-90: legs equal x; hypotenuse x√2.
  • 30-60-90: short leg x; long leg x√3; hypotenuse 2x.

Worked Examples

Example 1. 45-45-90, leg = 5. Hypotenuse? Solution. 5√2.

Example 2. 30-60-90, short leg = 4. Long leg? Hypotenuse? Solution. 4√3 and 8.

Example 3. Hypotenuse of a 30-60-90 is 12. Find each leg. Solution. Short leg = 6, long leg = 6√3.

Example 4. Diagonal of a square with side 7. Solution. 45-45-90 → 7√2.

Common Mistakes

  • Swapping the roles of short and long legs in 30-60-90.
  • Forgetting that the hypotenuse is the longest side.

Practice

  1. 45-45-90 with leg 9 — hypotenuse?
  2. 30-60-90 with short leg 7 — long leg and hypotenuse?
  3. Diagonal of a square with side 5√2.

Answers: 1. 9√2 | 2. 7√3 and 14 | 3. 10

17. Circle Theorems

Key Concepts

  • A central angle equals the arc it intercepts (degrees).
  • An inscribed angle is half the central angle (and half its intercepted arc).
  • An inscribed angle in a semicircle is 90°.
  • Tangent to a circle is perpendicular to the radius at the point of tangency.

Worked Examples

Example 1. Central angle 80°. Intercepted arc? Solution. 80°.

Example 2. Inscribed angle of 35° subtending an arc — arc measure? Solution. 2 × 35 = 70°.

Example 3. A right triangle inscribed in a circle. What does the hypotenuse pass through? Solution. The center of the circle (hypotenuse is a diameter).

Example 4. Two tangents from the same external point to a circle. Their lengths are: Solution. Equal.

Common Mistakes

  • Doubling instead of halving when converting between central and inscribed.
  • Assuming tangent and radius are parallel (they're perpendicular).

Practice

  1. Inscribed angle 50°. Arc?
  2. Central angle 120°. Inscribed angle on the major arc?
  3. Inscribed angle in a semicircle?

Answers: 1. 100° | 2. 60° | 3. 90°

18. Coordinate Geometry

Key Concepts

  • Distance: √[(x₂ − x₁)² + (y₂ − y₁)²].
  • Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
  • Slope: (y₂ − y₁)/(x₂ − x₁).
  • Parallel lines: equal slopes.
  • Perpendicular lines: slopes multiply to −1 (negative reciprocals).

Worked Examples

Example 1. Midpoint of (2, 4) and (8, 10). Solution. (5, 7).

Example 2. Distance between (1, 1) and (4, 5). Solution. √(9 + 16) = 5.

Example 3. Slope of (3, 6) and (5, 12). Solution. 6/2 = 3.

Example 4. Perpendicular slope to 2/3? Solution. −3/2.

Example 5. A triangle has vertices (0, 0), (4, 0), (4, 3). Right triangle? Solution. The vertex at (4, 0) has perpendicular legs along axes → Yes.

Common Mistakes

  • Adding instead of averaging for midpoint.
  • Forgetting the negative when finding the perpendicular slope.

Practice

  1. Midpoint of (−3, 4) and (5, −2).
  2. Distance between (−1, 0) and (2, 4).
  3. Line perpendicular to y = (1/4)x + 5 — slope?
  4. Are (1, 1), (4, 5), (7, 9) collinear?

Answers: 1. (1, 1) | 2. 5 | 3. −4 | 4. Yes (constant slope 4/3)

19. Sampling Methods and Bias

Key Concepts

  • Simple random: each member of the population has equal chance.
  • Stratified: divide into groups, sample from each.
  • Systematic: every nth member.
  • Convenience: easy to reach — usually biased.
  • Voluntary response: self-selected — biased.

Worked Examples

Example 1. A teacher polls students who stay after class — bias type? Solution. Convenience / voluntary bias.

Example 2. A pollster calls every 100th number from a phone book — sampling method? Solution. Systematic.

Example 3. A 1,000-student survey: 600 girls / 400 boys. Take 6 girls and 4 boys — sampling method? Solution. Stratified (proportional).

Common Mistakes

  • Treating any random selection as unbiased — choice of population matters too.

Practice

  1. A radio host asks listeners to call in — method? Bias?
  2. Every 50th item on an assembly line is checked. Method?
  3. A high-school survey samples 10% from each grade level. Method?

Answers: 1. Voluntary response (biased) | 2. Systematic | 3. Stratified

20. Normal Distribution

Key Concepts

A normal distribution is bell-shaped, symmetric, with mean μ in the center.

Empirical (68-95-99.7) rule:

  • ~68% within 1 standard deviation of the mean.
  • ~95% within 2.
  • ~99.7% within 3.

Worked Examples

Example 1. Mean = 70, SD = 5. About what % between 65 and 75? Solution. 1 SD on each side → 68%.

Example 2. Mean = 100, SD = 15. About what % above 130? Solution. 130 = mean + 2 SD. Right tail of 2 SD = 2.5%.

Example 3. Test scores N(80, 6). About what % between 68 and 92? Solution. ±2 SD → 95%.

Example 4. % below mean − 3 SD? Solution. 0.15%.

Common Mistakes

  • Confusing single-tail and double-tail percentages.
  • Misremembering the rule (68/95/99.7).

Practice

  1. N(50, 10). % between 40 and 60?
  2. N(120, 20). % above 160?
  3. N(0, 1). % between −1 and 1?

Answers: 1. ~68% | 2. ~2.5% | 3. ~68%

21. Expected Value

Key Concepts

Expected value E(X) = Σ x·P(x). It's the long-run average outcome.

Worked Examples

Example 1. A fair coin: win $5 for heads, lose $3 for tails. E? Solution. 0.5(5) + 0.5(−3) = $1.

Example 2. A die game: win $6 if you roll a 6, lose $1 otherwise. E? Solution. (1/6)(6) + (5/6)(−1) = 1 − 5/6 = $1/6 ≈ $0.17.

Example 3. Lottery ticket costs $2. 1/1000 chance of winning $1,000. E? Solution. (0.001)(1000) − (0.999)(2) = 1 − 1.998 = −$0.998.

Common Mistakes

  • Forgetting to subtract the cost of playing.
  • Mixing probability with payout when multiplying.

Practice

  1. Die game: win $5 if even, lose $2 if odd. E?
  2. Carnival game: 0.1 chance of $20 prize, 0.9 of nothing. Pay $3 to play. E?
  3. Two coins: win $10 if both heads, lose $2 otherwise. E?

Answers: 1. $1.50 | 2. −$1 | 3. $1 (0.25 × 10 + 0.75 × −2 = 2.5 − 1.5)

22. Linear Regression

Key Concepts

A linear regression line is the best-fit straight line for a set of data: y = a + bx.

The correlation coefficient r (between −1 and 1) measures linear association strength.

  • r near 1 → strong positive.
  • r near −1 → strong negative.
  • r near 0 → no linear pattern.

Use regression to interpolate (within the data range) — be careful extrapolating beyond.

Worked Examples

Example 1. A regression gives y = 4 + 2x. Predict y when x = 7. Solution. 4 + 14 = 18.

Example 2. r = −0.85 means strong negative linear association.

Example 3. y = 50 − 3x relates hours studied to mistakes. Each extra hour reduces mistakes by: Solution. 3.

Common Mistakes

  • Saying correlation = causation.
  • Extrapolating wildly past the data.

Practice

  1. r = 0.92 — interpret.
  2. y = 100 − 4x. Predict y at x = 10.
  3. y = 5 + 0.2x. Slope means each x adds...

Answers: 1. Strong positive linear association | 2. 60 | 3. 0.2 to y

That's Grade 9. Quadratics show up everywhere — factoring, formula, vertex, graphing. Master all four representations, plus solid algebra fundamentals, and you'll be ready for Grade 10.