Formulas

Standard Level

• Equations of a straight line: y=mx c; ax+by+d=0; y-y₁=m(x -x₁)

• Gradient formula: m=y₂-y_1×2-x₁

• Axis of symmetry of the graph of a quadratic function: f(x)=ax²+bx+c

• Axis of symmetry is x= -b/(2a)

Higher Level

• Logistic Function: f(x)=L/(1+Ce^-kx), where L, k, and C>0

Section A: Forms of Linear Lines

Linear Lines

• Refers to straight lines and can be expressed in various forms:

  • slope-intercept form 

  • point-slope form

  • standard form.

Slope-Intercept Form

• y = mx + c, where m is the gradient (slope) of the line and c is the y-intercept.

• Useful for its simplicity; allows easy identification of slope and y-intercept.

  • Example:y=(2/3)x+2

Point-Slope Form

• y - y₁=m(x -x₁), where (x₁, y₁) is a point on the line and m is the slope.

• useful when the y-intercept is not given, requiring a specific point on the line.

  • example: y-4=23(x-3)

Standard-Form

• Ax + By = C, where A, B, and C are constants, and A and B are not both zero.

• all terms must be on one side of the equation and integer coefficients.

  • example: -2x+3y-6=0

Things to Note

• Often, questions in IB exams may ask for the final answer in standard form, necessitating rearrangement from other forms.

• The process for converting from point-slope or slope-intercept to standard form involves expansion, grouping like terms, and sometimes multiplication to eliminate fractions.

• Both slope-intercept and standard forms represent the same equation, just arranged differently.

• Depending on the question's requirements, either form may be suitable for the final answer.

Section B: Gradients and Intercepts

Gradient (Slope)

• Represents the steepness of the line.

• Analogous to the slope of a hill; steepness determines the rate of change.

• Gradient (m) = rise/run    m=y2-y1x2-x1

• Δy represents the change in y and Δx represents the change in x between two points on the line.

Intercept Basics

• X-intercept: Where the line intersects the horizontal axis (x-axis).

• Y-intercept: Where the line intersects the vertical axis (y-axis).

• A straight line will have only one x-intercept and one y-intercept.

• Positive gradient: Line slopes upwards from left to right. Negative gradient: The line slopes downwards from left to right.

Section C: Parallel and Perpendicular Gradients

Parallel Lines

• Lines that are always equidistant and never intersect, resembling train tracks.

• The gradients of parallel lines are equal to each other.

• Example: If Line 1 has a gradient of 2, then Line 2, parallel to Line 1, also has a gradient of 2.

Perpendicular Lines

• Lines that intersect at a right angle (90 degrees), forming a "T" shape.

• The gradients of perpendicular lines are negative reciprocals of each other.

• Negative reciprocal means taking the negative of the reciprocal of the gradient of one line.

• Example: If Line 1 has a gradient of 2, then Line 2, perpendicular to Line 1, has a gradient of -1/2.

Section D: Perpendicular Bisectors

Perpendicular Bisector

• Application primarily appears in Voronoi diagrams.

• Definition: A line segment that cuts another line segment into two equal parts at a right angle.

• The midpoint of the line segment defines the point through which the perpendicular bisector passes.

Example:

• Given a line segment AB, find the equation of the perpendicular bisector.

  • Step 1: Find the midpoint of AB using the midpoint formula.

  • Step 2: Determine the gradient of AB using the gradient formula.

  • Step 3: Apply the concept of perpendicular lines to find the gradient of the perpendicular bisector.

  • Step 4: Use the point-slope form of a linear equation to find the equation of the perpendicular bisector.

• Example Solution:

  • Given line segment AB with A(-1,1) and B(3,3).

  • Step 1: Midpoint = (1,2)

  • Step 2: Gradient of AB = ½

  • Step 3: Gradient of perpendicular bisector = -1 / (1/2) = -2

  • Step 4: Equation of perpendicular bisector = y = -2x + 4

Section E: Types of Functions

• A function is described as a mathematical rule denoted by f(x)=something

• It represents the relationship between two variables, where:

  • x is the input (independent variable) and

  • y is the output (dependent variable).

  • Example:

    • Throwing a rock down a ditch 

    • Time (input) influences height (output).

    • Modelled with a linear function:

      • H(t)=10a+50

Types of Functions

• Linear Functions: Have a constant rate of change, forming straight lines.

• Quadratic Functions: Involve squares of the variable, often representing parabolic curves.

• Exponential Functions: Involve a constant raised to a variable power, used for growth or decay processes.

• Logarithmic Functions: The inverse of exponential functions, used to model decreasing growth rates.

• Sinusoidal Functions: Represent periodic oscillations, such as waves or tides.

• Logistic Functions: Combine exponential and logarithmic functions, useful for modelling growth that levels off.

• Piecewise Functions: Combinations of different functions defined over specific intervals, useful for modeling complex scenarios with changing behaviors.

HL Only: Domain & Range, Composite, Inverse

Function Representation

• Functions can be represented in various ways, such as:

• (x)=x₂−2, f:x↦x₂−2 f:x↦x₂−2, or y=x₂−2y=x₂−2.

• All these representations convey the same meaning, depending on the context or textbook used.

Domain and Range

• Domain refers to the set of possible input values (X), while range refers to the set of possible output values (Y).

• For example, for f(x)=x₂−2

  • Domain: x∈R, where x∈R means all real numbers

  • Range: y ≥-2 (all real numbers greater than or equal to -2).

Composite Functions

• Composite functions, denoted as

  • f∘g, or g∘f, involve applying one function within another function.

• For example, if f(x)=x₂−2 and g(x)=x+3

  • f∘g: Replace x in f(x) with g(x), i.e., (x+3)2−2

  • g∘f Replace x in g(x) with f(x), i.e., (x₂−2)+3

Inverse Functions

• The inverse function, denoted as f-1, reverses the action of the original function.

• To find the inverse of a function:

  • Rewrite the function in the form y=f(x)

  • Swap x and y and solve for y.

  • The resulting equation is the inverse function.

• For example, if f(x)=x2−2

  • Rewrite as y=x₂−2

  • Swap x and y to get x=y₂−2

  • Solve for y to get f-1(x)=x+2

Understanding Functions

• Composite functions and inverse functions have deeper meanings and are essential in understanding the behavior and relationships between functions.

• Inverse functions represent reflections across the line y=x, while composite functions involve applying one function's output as another function's input.

HL Only: Transformation of Functions

Functions Representation

• Functions are often represented as f(x), where x is the input variable and f(x) is the output.

• Knowing a coordinate on the function, such as (2,2), helps understand its behavior.

Translation (Shift)

• Translations involve shifting the function horizontally or vertically.

• Horizontal shift: f(x±a)

  • f(x±a) shifts a units left or right.

• Vertical shift: f(x)±a

  • f(x)±a shifts a units up or down.

• Remember that signs inside and outside the function bracket indicate horizontal and vertical shifts, respectively.

Stretch

• Stretches involve making the function skinnier or wider.

• Vertical stretch: a⋅f(x)

  • a⋅f(x) multiplies all y values by a.

• Horizontal stretch: f(ax)

  • f(ax) stretches the function horizontally by a factor of 1a

  • Horizontal stretch by 1a means compressing if a>1 or stretching if 0<a<1.

Reflection

• Reflections involve flipping the function over the x-axis or y-axis.

• Reflection in the x-axis: −f(x)

  • −f(x) reflects all y values vertically.

• Reflection in the y-axis: f(−x)

  • f(−x) reflects all

  • x values horizontally.

Summary

• Horizontal shifts affect x values inside the function.

• Vertical shifts affect y values outside the function.

• Stretches scale the function vertically or horizontally.

• Reflections flip the function over the x-axis or y-axis.

Plotting Functions and Analysis Tools

Plotting a Quadratic Function

• Example: Height of a cannonball over time.

  • Enter function:

    • −2x²+20x+8

  • Adjust zoom settings for accurate visualization. (I suggest [ZOOM] -> 0)

Analysis Tools:

• Maximum Height:

  • Use "Maximum" tool.

  • After 5 seconds, cannonball reaches 58 meters.

• Time to Reach Ground:

  • Utilize "Zero" tool by pressing [2ND] + [TRACE] + 2

  • “Left bound?” -> Move “blinky” to left side of line from where it touches at zero

  • “Right bound?” -> move “blinky” to right side from where it touches at zero

  • “Guess?” -> place “blinky” on top of where it touches zero

  • Result: 10.38 (round up by three sig-figs) -> 10.4 

  • Interpretation: Cannonball touches ground at 10.4 seconds 

• Initial Height:

  • Plug in 0 for function; in most cases, getting an initial height can be found by isolating the constant alone.

  • In context: initial height is 8 meters.

• Time to Reach Specific Height:

  • Plot horizontal line (e.g., at 40 meters).

  • Use "Intersection" tool.

  • Cannonball reaches 40 meters in 2 seconds.

Tips: Numeric Solver

• Numeric Solver (N Solve) is a valuable function on TI-84 for solving equations quickly and accurately.

• It's particularly essential for topics like exponential, logarithmic, quadratic, and cubic functions.

Accessing N Solve

• Access the function by navigating to [MATH] > [C:Numeric Solver] 

Entering Equation

• Input the equation directly into the calculator, ensuring correct syntax and using 'X' as the variable.

Specifying Bounds

• Specify appropriate bounds to narrow down the search range for the solution.

• Bounds should be close to the expected solution for faster convergence.

Initiating Calculation

• Press [GRAPH] button when ready to start “guessing”

• Input any value for X (as long as it fits in context to problem)

• Result will be displayed in the same area you input a value in

Viewing Result

• The calculator displays the approximate solution once the calculation is complete.

• REMEMBER TO USE THREE SIG-FIGS

Considerations

• Numeric Solver provides the left-hand solution for equations with multiple solutions.

• For equations with multiple solutions or complex functions, consider using other analysis tools like plotting functions and intersection analysis.