Mathematics
Grade 12
15 min
Dilations of functions
Dilations of functions
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1
Introduction & Learning Objectives
Learning Objectives
Define vertical and horizontal dilations and identify their corresponding parameters in a function's equation.
Describe the specific transformation (stretch or compression) and its scale factor, given a function in the form y = af(kx).
Write the equation of a new function that results from applying a specified vertical or horizontal dilation to a parent function.
Accurately graph a dilated function by transforming key points from its parent function.
Determine the equation of a dilated function by analyzing its graph in relation to the parent function's graph.
Explain how dilations affect key properties of functions, such as amplitude, period, and intercepts.
Apply mapping notation (x, y) -> (x/k, ay) to determine the coordinates of points on...
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Key Concepts & Vocabulary
TermDefinitionExample
DilationA transformation that stretches or compresses a graph. Dilations change the size and shape of the graph but do not change its fundamental form.Transforming the parabola y = x^2 into the narrower parabola y = 3x^2 is a vertical dilation.
Vertical DilationA dilation that stretches or compresses the graph away from or towards the x-axis. It is controlled by a parameter that multiplies the entire function.Given f(x) = |x|, the function g(x) = 0.5|x| represents a vertical compression by a factor of 0.5.
Horizontal DilationA dilation that stretches or compresses the graph away from or towards the y-axis. It is controlled by a parameter that multiplies the input variable 'x'.Given f(x) = sin(x), the function g(x) = sin(2x) represents a horizontal compressi...
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Core Formulas
Vertical Dilation
y = af(x)
This transformation affects the y-coordinates. If |a| > 1, it's a vertical stretch by a factor of 'a'. If 0 < |a| < 1, it's a vertical compression by a factor of 'a'. If a < 0, the dilation is combined with a reflection across the x-axis. The mapping is (x, y) -> (x, ay).
Horizontal Dilation
y = f(kx)
This transformation affects the x-coordinates. If |k| > 1, it's a horizontal compression by a factor of '1/k'. If 0 < |k| < 1, it's a horizontal stretch by a factor of '1/k'. If k < 0, the dilation is combined with a reflection across the y-axis. The mapping is (x, y) -> (x/k, y).
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Challenging
Given f(x) = x^2 - 9, which has roots at x = ±3. If g(x) = 0.5f(3x), what are the roots of g(x)?
A.x = ±1.5
B.x = ±6
C.x = ±9
D.x = ±1
Challenging
For a function of the form f(x) = x^n, under what condition is a vertical stretch by a factor of c^n (where c > 1) equivalent to a horizontal compression by a factor of 1/c?
A.Only when n = 1
B.For all n > 0
C.Only when n = 0
D.This is never possible
Challenging
The definite integral of a continuous function f(x) from x=1 to x=5 represents the net area under its curve. If ∫[1 to 5] f(x) dx = 10, what is the value of ∫[1 to 5] 3f(x) dx?
A.30
B.10/3
C.13
D.10
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