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5. Functions Grade 10

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Functions

A function is a mathematical relationship between two variables, where every input

variable has one output variable. The x-variable is known as the input or independent variable, because its value

can be chosen freely. The calculated y-variable is known as the output or dependent variable, because

its value depends on the chosen input value. An asymptote is a straight line, which the graph of a function will approach, but

never touch. A graph is said to be continuous if there are no breaks in the graph.

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Different notation Set notation :

Interval notation:

Function notation:

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Domain and range Domain: the set of all independent x-values for which there is one dependent

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y-value according to that function. Range: the set of all dependent y-values which can be obtained using an independent

â—?

x-value.

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Linear functions y = f(x) = x Basic equation: y = x

Intercept: (0; 0)

Domain: x ∈ R

Range: f(x) ∈ R

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Linear functions y = f(x) = mx + c Equations of the form:

y = mx + c

Intercepts:

Let x = 0,

(0; c)

Let y = 0,

(x; 0)

Domain: x ∈ R

Range: f(x) ∈ R

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Linear functions y = mx + c The gradient of a line is determined by the ratio of vertical change to horizontal

â—?

change: The effects of m and c:

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Linear functions y = f(x) = mx + c Dual intercept method:

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1. Determine y-intercept; let x = 0 and solve. 2. Determine x-intercept; let y = 0 and solve. 3. Use the two intercepts to sketch the graph. â—?

Gradient and y-intercept method:

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1. Determine y-intercept; let x = 0 and solve. 2. Determine gradient, equation must be in standard form y = mx + c 3. Use the y-intercept and gradient to determine second point on the line.

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Quadratic functions y = f(x) = x2 Basic equation: y = x2

Intercepts: (0; 0)

Domain: x ∈ R

Range: {y : y ∈ R, y ≥ 0}

Axis of symmetry: x = 0

Turning point: (0; 0)

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Quadratic functions y = f(x) = ax2 + q Equations of the form:

y = ax2 + q

Intercepts:

Let x = 0,

(0; q)

Let y = 0,

0 = ax2 + q and solve for x

Domain: x ∈ R

Range: if a > 0, f(x) > q

if a < 0, f(x) < q Axis of symmetry: x = 0

Turning point: (0; q)

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Quadratic functions y = ax2 + q The effects of a and q:

â—?

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Hyperbolic functions Basic equation:

Intercepts: none

Domain: {x : x ∈ R, x ≠ 0}

Range: {y : y ∈ R, y ≠ 0}

Axis of symmetry: y = x and y = -x

Asymptotes:

horizontal y = 0 vertical x = 0

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Hyperbolic functions Equations of the form:

Intercepts:

Let x = 0,

no y-intercept

Let y = 0,

and solve for x

Domain: {x : x ∈ R, x ≠ 0}

Range: {y : y ∈ R, y ≠ q}

Axis of symmetry: y = x + q and y = -x + q

Asymptotes:

horizontal y = q vertical x = 0

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Hyperbolic functions The effects of a and q:

â—?

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Exponential functions y = f(x) = bx Basic equation: y = bx, b > 0 and b ≠ 1.

Intercepts:

Let x = 0,

(0; 1)

Let y = 0,

no x-intercept

Domain: x ∈ R

Range: {y : y ∈ R, y > 0}

Asymptote: y = 0

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Exponential functions y = f(x) = abx + q Equation of the form: y = abx + q

Intercepts:

Let x = 0,

(0; a + q)

Let y = 0,

and solve for x

Domain: x ∈ R

Range: if a > 0, f(x) > q

if a < 0, f(x) < q Axis of symmetry: y = x + q and y = -x + q

Asymptotes:

horizontal y = q

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Exponential functions y = f(x) = abx + q The effects of a and q:

â—?

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Sine functions y = f(x) = sin θ Basic equation: y = sin θ

Domain: [ 00 ; 3600 ]

Range: [−1; 1]

x-intercepts: (00 ; 0), (1800 ; 0), (3600 ; 0)

y-intercept: (00 ; 0)

Maximum turning point: (900 ; 1)

Minimum turning point: (2700 ; −1)

Period: 3600

Amplitude: 1

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Sine functions y = f(x) = a sin θ + q Equation of the form: y = a sin θ + q

Domain: [ 00 ; 3600 ]

Range: if a > 0, f (θ) ∈ [−a + q, a + q]

if a < 0, f (θ) ∈ [a + q, −a + q] Period: 3600

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Sine functions y = f(x) = a sin θ + q The effects of a and q

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Cosine functions y = f(x) = cos θ Basic equation: y = cos θ

Range: [−1; 1]

x-intercepts: (90◦ ; 0), (270◦ ; 0)

y-intercept: (0◦ ; 1)

Maximum turning points: (0◦ ; 1), (360◦ ; 1)

Minimum turning point: (180◦ ; −1)

Period: 3600

Amplitude: 1

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Cosine functions y = f(x) = a cos θ + q Equation of the form: y = a cos θ + q

Domain: [ 00 ; 3600 ]

Range: if a > 0, f (θ) ∈ [−a + q, a + q]

if a < 0, f (θ) ∈ [a + q, −a + q] Period: 3600

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Cosine functions y = f(x) = a cos θ + q The effects of a and q:

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Comparison of graphs of y = sin θ and y = cos θ

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Tangent functions y = f(x) = tan θ Basic equation: y = tan θ

Asymptotes: the lines θ = 90◦ and θ = 270◦

Period: 180◦

Domain: {θ : 0◦ ≤ θ ≤ 360◦ , θ ≠ 90◦ ; 270◦ }

Range: f (θ) ∈ R

x-intercepts: (0◦ ; 0), (180◦ ; 0), (360◦ ; 0)

y-intercept: (0◦ ; 0)

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Tangent functions y = f(x) = a tan θ + q Equation of the form: y = a tan θ + q

Asymptotes: the lines θ = 90◦ and θ = 270◦

Period: 180◦

Domain: {θ : 0◦ ≤ θ ≤ 360◦ , θ ≠ 90◦ ; 270◦ }

Range: f(θ) ∈ R

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Tangent functions y = f(x) = a tan θ + q The effects of a and q:

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For more practice or to ask an expert for help on this section see: www.everythingmaths.co.za Shortcode: EMAAM

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