Mathematics
Grade 10
15 min
Transformations of quadratic functions
Transformations of quadratic functions
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1
Introduction & Learning Objectives
Learning Objectives
Identify the parameters a, h, and k in the vertex form of a quadratic function.
Describe the effects of vertical stretches, compressions, and reflections on the graph of y = x^2.
Describe the effects of horizontal and vertical translations on the graph of y = x^2.
Graph a quadratic function in vertex form by applying a sequence of transformations to the parent function.
Write the equation of a transformed quadratic function given its graph or a description of the transformations.
Determine the vertex, axis of symmetry, and direction of opening from an equation in vertex form.
Ever wonder how a basketball player makes a perfect shot or how a bridge arch is designed? 🏀 The secret lies in understanding the elegant, predictable curves of parabolas!
In this...
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Key Concepts & Vocabulary
TermDefinitionExample
Parent FunctionThe simplest form of a function in a family. For quadratics, this is the function to which all transformations are applied.The parent quadratic function is f(x) = x^2. Its graph is a parabola with a vertex at the origin (0, 0).
VertexThe highest or lowest point on a parabola. It is the point where the parabola changes direction.For the parabola y = (x - 2)^2 + 3, the vertex is at the point (2, 3).
Axis of SymmetryA vertical line that passes through the vertex and divides the parabola into two mirror-image halves.For the parabola y = (x - 2)^2 + 3, the axis of symmetry is the vertical line x = 2.
TranslationA transformation that 'slides' every point of a figure or a graph the same distance in the same direction without changing its size or ori...
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Core Formulas
Vertex Form of a Quadratic Function
f(x) = a(x - h)^2 + k
This is the key formula for understanding transformations. The parameters 'a', 'h', and 'k' tell you exactly how the parent function f(x) = x^2 has been transformed. The vertex of the parabola is located at the point (h, k).
The Role of Parameter 'a'
y = a x^2
The parameter 'a' controls vertical dilation and reflection. If a < 0, the parabola is reflected across the x-axis (opens downward). If |a| > 1, the parabola is vertically stretched (appears narrower). If 0 < |a| < 1, the parabola is vertically compressed (appears wider).
The Role of Parameters 'h' and 'k'
(x - h) and + k
The parameters 'h' and 'k' contr...
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Challenging
What is the y-intercept of the function f(x) = -2(x - 3)^2 + 10?
A.(0, 10)
B.(0, -8)
C.(3, 10)
D.(0, -26)
Challenging
The graph of f(x) = a(x - h)^2 + k is transformed to create g(x) = -a(x - (h+2))^2 + (k-3). How does the vertex of g(x) relate to the vertex of f(x)?
A.It is shifted 2 units left and 3 units up.
B.It is shifted 2 units right and 3 units down.
C.It is shifted 2 units left and 3 units down.
D.It is shifted 2 units right and 3 units up.
Challenging
A parabola opens downwards, and its vertex is in Quadrant IV. What must be true about the parameters a, h, and k in its equation f(x) = a(x - h)^2 + k?
A.a > 0, h > 0, k < 0
B.a < 0, h < 0, k < 0
C.a < 0, h > 0, k > 0
D.a < 0, h > 0, k < 0
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