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5. Functions Grade 10
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Functions
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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
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can be chosen freely. The calculated y-variable is known as the output or dependent variable, because
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its value depends on the chosen input value. An asymptote is a straight line, which the graph of a function will approach, but
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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 :
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Interval notation:
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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
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x-value.
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Linear functions y = f(x) = x Basic equation: y = x
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Intercept: (0; 0)
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Domain: x ∈ R
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Range: f(x) ∈ R
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Linear functions y = f(x) = mx + c Equations of the form:
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y = mx + c
Intercepts:
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Let x = 0,
(0; c)
Let y = 0,
(x; 0)
Domain: x ∈ R
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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
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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
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Intercepts: (0; 0)
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Domain: x ∈ R
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Range: {y : y ∈ R, y ≥ 0}
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Axis of symmetry: x = 0
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Turning point: (0; 0)
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Quadratic functions y = f(x) = ax2 + q Equations of the form:
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y = ax2 + q
Intercepts:
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Let x = 0,
(0; q)
Let y = 0,
0 = ax2 + q and solve for x
Domain: x ∈ R
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Range: if a > 0, f(x) > q
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if a < 0, f(x) < q Axis of symmetry: x = 0
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Turning point: (0; q)
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Quadratic functions y = ax2 + q The effects of a and q:
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Hyperbolic functions Basic equation:
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Intercepts: none
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Domain: {x : x ∈ R, x ≠ 0}
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Range: {y : y ∈ R, y ≠ 0}
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Axis of symmetry: y = x and y = -x
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Asymptotes:
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horizontal y = 0 vertical x = 0
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Hyperbolic functions Equations of the form:
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Intercepts:
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Let x = 0,
no y-intercept
Let y = 0,
and solve for x
Domain: {x : x ∈ R, x ≠ 0}
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Range: {y : y ∈ R, y ≠ q}
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Axis of symmetry: y = x + q and y = -x + q
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Asymptotes:
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horizontal y = q vertical x = 0
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Hyperbolic functions The effects of a and q:
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Exponential functions y = f(x) = bx Basic equation: y = bx, b > 0 and b ≠ 1.
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Intercepts:
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Let x = 0,
(0; 1)
Let y = 0,
no x-intercept
Domain: x ∈ R
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Range: {y : y ∈ R, y > 0}
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Asymptote: y = 0
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Exponential functions y = f(x) = abx + q Equation of the form: y = abx + q
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Intercepts:
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Let x = 0,
(0; a + q)
Let y = 0,
and solve for x
Domain: x ∈ R
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Range: if a > 0, f(x) > q
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if a < 0, f(x) < q Axis of symmetry: y = x + q and y = -x + q
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Asymptotes:
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horizontal y = q
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Exponential functions y = f(x) = abx + q The effects of a and q:
â&#x2014;?
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Sine functions y = f(x) = sin θ Basic equation: y = sin θ
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Domain: [ 00 ; 3600 ]
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Range: [−1; 1]
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x-intercepts: (00 ; 0), (1800 ; 0), (3600 ; 0)
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y-intercept: (00 ; 0)
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Maximum turning point: (900 ; 1)
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Minimum turning point: (2700 ; −1)
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Period: 3600
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Amplitude: 1
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Sine functions y = f(x) = a sin θ + q Equation of the form: y = a sin θ + q
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Domain: [ 00 ; 3600 ]
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Range: if a > 0, f (θ) ∈ [−a + q, a + q]
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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
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Range: [−1; 1]
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x-intercepts: (90◦ ; 0), (270◦ ; 0)
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y-intercept: (0◦ ; 1)
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Maximum turning points: (0◦ ; 1), (360◦ ; 1)
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Minimum turning point: (180◦ ; −1)
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Period: 3600
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Amplitude: 1
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Cosine functions y = f(x) = a cos θ + q Equation of the form: y = a cos θ + q
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Domain: [ 00 ; 3600 ]
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Range: if a > 0, f (θ) ∈ [−a + q, a + q]
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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
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Asymptotes: the lines θ = 90◦ and θ = 270◦
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Period: 180◦
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Domain: {θ : 0◦ ≤ θ ≤ 360◦ , θ ≠ 90◦ ; 270◦ }
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Range: f (θ) ∈ R
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x-intercepts: (0◦ ; 0), (180◦ ; 0), (360◦ ; 0)
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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
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Asymptotes: the lines θ = 90◦ and θ = 270◦
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Period: 180◦
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Domain: {θ : 0◦ ≤ θ ≤ 360◦ , θ ≠ 90◦ ; 270◦ }
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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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