Flashcards · 4024 · Topic 2

Algebra and graphs flashcards

61 cards on Topic 2, subtopics 2.1 to 2.12. Pick a subtopic or study them all.

Study mode needs JavaScript. Every card is listed below, so you can still cover the answers and test yourself.

All 61 cards

For a quick read-through or printing. Cover the right-hand side to test yourself.

2.1 Introduction to algebra

Like terms: Terms with exactly the same letters raised to exactly the same powers. 3a² and 5a² are like terms; 3a² and 3a are not.
Find the value of 3x² − 2x when x = −2: 3 × (−2)² − 2 × (−2) = 3 × 4 + 4 = 16. Square the negative number: (−2)² = 4.
Simplify 2a² + 3ab − 1 + 5a² − 9ab + 4: a² terms: 7a². ab terms: −6ab. Numbers: 3. Answer: 7a² − 6ab + 3
Write an expression for the product of two consecutive even numbers: n(n + 2), where n is an even number. Letters stand for general numbers, so the expression works for any case.

2.2 Algebraic manipulation

Expand (x + a)²: x² + 2ax + a² Trap: (x + a)² is not x² + a². The middle term 2ax is compulsory.
Difference of two squares: x² − a² = (x + a)(x − a) Example: x² − 25 = (x + 5)(x − 5).
Factorising: what order do you check in?: 1. Common factor first, always. 2. Difference of two squares. 3. Quadratic trinomial: two numbers that multiply to ac and add to b. 4. Grouping, for four terms.
Factorise fully 2x³ + 10x² + 12x: Common factor first: 2x(x² + 5x + 6) Then the quadratic: 2x(x + 2)(x + 3)
Complete the square: x² + 8x + 3: (x + 4)² − 16 + 3 = (x + 4)² − 13 General: x² + bx + c = (x + b/2)² − (b/2)² + c. Check by expanding back out.
Factorise by grouping: ax + ay + bx + by: a(x + y) + b(x + y) = (a + b)(x + y) Pair terms that share a factor, then take out the common bracket.

2.3 Algebraic fractions

How do you simplify an algebraic fraction?: Factorise the top and the bottom, then cancel common factors. Never cancel individual terms: in (x + 2)/(x + 4) nothing cancels.
Simplify (x² − 9)/(x² + 4x + 3): (x + 3)(x − 3) / ((x + 3)(x + 1)) = (x − 3)/(x + 1)
Simplify x/3 + (x − 4)/2: Common denominator 6: 2x/6 + 3(x − 4)/6 = (2x + 3x − 12)/6 = (5x − 12)/6
How do you divide algebraic fractions? (3a/4 ÷ 9a/10): Multiply by the reciprocal of the second fraction: 3a/4 × 10/(9a) = 30a/(36a) = 5/6

2.4 Indices II

Index laws for multiplying, dividing and raising a power: aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ The base must be the same.
What is 2⁻³?: 1/2³ = 1/8. A negative index means a reciprocal, not a negative number (it is not −8).
Evaluate 27^(2/3): Root first, then power: (∛27)² = 3² = 9.
Simplify (2x³)⁴: 2⁴ × x¹² = 16x¹². Raise the number and the letter to the power 4.
Solve 5^(x+1) = 25ˣ: Write both sides with base 5: 5^(x+1) = 5^(2x). Equate the indices: x + 1 = 2x, so x = 1.

2.5 Equations

Simultaneous equations: when is elimination a good method?: When both equations are linear. Make the coefficients of one letter match, then add or subtract the equations.
Simultaneous equations: when is substitution a good method?: When one equation is already in the form y = … (or can easily be rearranged into it). Substitute it into the other equation.
The quadratic formula: For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a) Rearrange to = 0 first. If b = −5, then −b = +5.
What does the discriminant b² − 4ac tell you?: Positive: two real roots. Zero: one repeated root. Negative: no real roots.
Changing the subject when the letter appears twice: make x the subject of ax + b = cx + d: Collect the x terms on one side: ax − cx = d − b. Factorise: x(a − c) = d − b. Divide: x = (d − b)/(a − c).
Solving an equation with x in a denominator: what must you check at the end?: Reject any solution that would make an original denominator zero.

2.6 Inequalities

When does an inequality sign reverse?: When you multiply or divide both sides by a negative number. Example: −3x > 12 gives x < −4. Adding and subtracting never reverse it.
Number line: open circle or filled circle?: Open circle: < or > (the end value is not included). Filled circle: ≤ or ≥ (the end value is included).
Graph of an inequality: dashed or solid boundary line?: Dashed: < or > (points on the line are not included). Solid: ≤ or ≥ (points on the line are included).
Solve −5 < 2x + 1 ≤ 9 and list the integer solutions: Subtract 1 from each part: −6 < 2x ≤ 8. Divide by 2: −3 < x ≤ 4. Integers: −2, −1, 0, 1, 2, 3, 4 (−3 is not included, 4 is).
How do you decide which side of a line satisfies an inequality?: Test a point that is not on the line, usually (0, 0). If it makes the inequality true, that side is the required region.

2.7 Sequences

How do you tell a linear, a quadratic and an exponential sequence apart?: Constant first difference: linear. Constant second difference: quadratic. Constant ratio between terms: exponential.
Find the nth term of 5, 8, 11, 14, …: Difference 3, so start with 3n: 3, 6, 9, 12. Each term is 2 more, so the nth term is 3n + 2.
Find the nth term of 3, 8, 15, 24, …: Second difference 2, so a = 2 ÷ 2 = 1: start with n². Terms − n²: 2, 4, 6, 8, which is 2n. nth term = n² + 2n.
Find the nth term of 2, 6, 18, 54, …: Constant ratio r = 3 and first term a = 2. nth term = arⁿ⁻¹ = 2 × 3ⁿ⁻¹.
Triangular numbers: 1, 3, 6, 10, 15, … The differences go up by 1 each time.

2.8 Proportion

y is directly proportional to x: equation and graph: y = kx, where k is a constant. The graph is a straight line through the origin. If x doubles, y doubles.
y is inversely proportional to x: equation and effect: y = k/x. If x doubles, y halves. The product xy stays constant.
Method for a proportion question: 1. Write the equation with k (for example y = kx²). 2. Substitute the given pair to find k. 3. Rewrite the equation with that k. 4. Use it to answer the question.
y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.: 45 = k × 9, so k = 5 and y = 5x². When x = 5: y = 5 × 25 = 125.

2.9 Graphs in practical situations

What does the gradient of a distance–time graph show?: The speed. A horizontal line means the object is stationary.
Speed–time graph: what do the gradient and the area under the graph show?: Gradient: acceleration. Area under the graph: distance travelled. A horizontal line means constant speed, not stopped.
A car goes from 0 to 15 m/s in 6 s at a steady rate. Find the acceleration and the distance.: Acceleration = 15 ÷ 6 = 2.5 m/s². Distance = area of triangle = ½ × 6 × 15 = 45 m.
How do you change km/h into m/s?: Divide by 3.6 (1 km/h = 1000 m ÷ 3600 s). Example: 72 km/h = 72 ÷ 3.6 = 20 m/s.
When does a conversion graph pass through the origin?: Only when the two quantities are directly proportional (for example miles and kilometres). The Celsius–Fahrenheit line does not, because 0 °C = 32 °F.

2.10 Graphs of functions

Shape of y = ax² + bx + c: A parabola with one turning point. It opens upwards (a minimum) if a > 0 and downwards (a maximum) if a < 0.
Shape of y = k/x: A hyperbola with two separate branches. The x-axis and y-axis are asymptotes: the curve gets closer and closer to them but never meets them.
Exponential graph y = abˣ + c: growth or decay?: Growth when b > 1; decay when 0 < b < 1 (for a > 0). The curve approaches the horizontal asymptote y = c but never reaches it.
How do you solve f(x) = g(x) using graphs?: Draw both graphs and read the x-coordinates of the points where they intersect. The answer is the x-values, not the coordinate pairs.
How do you find the gradient of a curve at a point?: Draw a tangent to the curve at that point, then find the gradient of the tangent using a large triangle.
The curve y = x² − 2x − 3 is drawn. Which line do you draw to solve x² − 3x − 1 = 0?: Rearrange so one side is the drawn curve: x² − 2x − 3 = x − 2. Draw y = x − 2 and read the x-coordinates where it meets the curve.

2.11 Sketching curves

Which key features should a sketch of a curve show?: The y-intercept (set x = 0), the x-intercepts or roots (set y = 0), any turning points, and any asymptotes, all labelled. It need not be to scale.
Turning point from completed square form y = a(x + p)² + q: Turning point at (−p, q). The line of symmetry is x = −p.
Sketch facts for y = x² − 2x − 3: y-intercept −3. Roots: (x + 1)(x − 3) = 0, so x = −1 and x = 3. Line of symmetry halfway between the roots, x = 1, so the minimum point is (1, −4).
Shape of a cubic y = ax³ + …: If a > 0 the curve rises from bottom left to top right; if a < 0 it falls from top left to bottom right. It has either two turning points or none.
What tells you the shape of a graph before you plot anything?: The highest power of x: 1 gives a straight line, 2 a parabola, 3 a cubic; x in a denominator (k/x) gives a hyperbola; x as a power gives an exponential.

2.12 Functions

Domain of a function: The set of inputs (x values) allowed.
Range of a function: The set of outputs (y values) produced.
f(x) = 2x + 1 and g(x) = x². Find fg(x).: fg(x) means do g first, then f: fg(x) = f(x²) = 2x² + 1. (gf(x) = (2x + 1)², which is different.)
How do you find an inverse function? (f(x) = 3x − 5): 1. Write y = 3x − 5. 2. Swap x and y: x = 3y − 5. 3. Make y the subject: y = (x + 5)/3. 4. f⁻¹(x) = (x + 5)/3.
What does f⁻¹(x) mean, and how is its graph related to y = f(x)?: The inverse function, which undoes f. It is not 1/f(x). Its graph is the reflection of y = f(x) in the line y = x.
Which value must be excluded from the domain of g(x) = 1/(x − 2)?: x = 2, because it makes the denominator zero and division by zero is undefined.