2.6.1
Properties of Functions
Properties of Functions
Properties of Functions
By defining certain properties of functions, we can get a sense of how they behave for different values of their input.
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Vertices
Vertices
- The vertex of a parabola is its turning point.
- This vertex can either be a maximum or a minimum.
- A “happy” parabola (of form ) has a vertex that is a minimum.
- A “sad” parabola (of form ) has a vertex that is a maximum.
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Axis of symmetry
Axis of symmetry
- The axis of symmetry is the line that cuts the parabola into two halves which are mirror images of each other.
- If the parabola is vertical, the equation of the axis of symmetry has the form , where is the -coordinate of the vertex.
![Illustrative background for Axis of symmetry](https://image-v2.cdn.app.senecalearning.com/2020-01/5e3b295d-865f-406f-8419-d780ecfa78ee/functions-axis-symmetry,h_400,q_80,w_640.png)
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Axis of symmetry
Axis of symmetry
- We can find the value of by looking at the general form of a quadratic equation .
- Completing the square on this expression gives:
- The -coordinate of the vertex is .
- So the axis of symmetry is given by the equation
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Average rate of change
Average rate of change
- A rate of change describes how an output quantity changes relative to the change in the input quantity.
- The units on a rate of change are “output units per input units.”
- To find the average rate of change of a function, we divide the change in the output value by the change in the input value.
- Average rate of change =
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Increasing or decreasing
Increasing or decreasing
- A function is increasing on an interval if the function values increase as the input values increase within that interval.
- A function is decreasing on an interval if the function values decrease as the input values decrease within that interval.
- The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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