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

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

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24 Terms

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Linear Lines

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

    • slope-intercept form 

    • point-slope form

    • standard form.

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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

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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

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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.

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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.

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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.

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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.

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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

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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

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Linear Functions

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

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Quadratic Functions

Involve squares of the variable, often representing parabolic curves.

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Exponential Functions

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

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Logarithmic Functions

The inverse of exponential functions, used to model decreasing growth rates

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Sinusoidal Functions

Represent periodic oscillations, such as waves or tides.

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Logistic Functions

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

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Piecewise Functions

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

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Function Representation

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

  • (x)=x2−2, f:x↦x2−2 f:xx2−2, or y=x2−2y=x2−2.

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

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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).

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Composite Functions

  • Composite functions, denoted as

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

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

    • fg: 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

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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

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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.

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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(xa shifts a units up or down.

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

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Stretch

  • Stretches involve making the function skinnier or wider.

  • Vertical stretch: a⋅f(x)

    • af(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.

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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.