IB Applications & Interpretations Topic 3 - Functions (SL and HL)

Definitions for the IB Applications & Interpretations Topic 3 - Functions (SL and HL).

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

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

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.

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.

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

Function

• 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

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.

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

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.